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Good search terms to look for are:
You’ll find a number of Suttas that talk of “practicing walking and sitting meditation [at different times of the day], purifying their mind from obstacles”. Others report the Buddha or other mendicants practicing walking meditation.
I suspect that the speaker you refer to is talking about the commentaries or even later practice traditions like the Mahasi tradition.
I’m not aware of any early discourses that differentiate between the two postures regarding potential for calm or insight. Usually they’re treated as just two different postures through which one maintains continuity in practice. For example, the stock description of devotion to wakefulness, as part of the gradual training:
And how is a bhikkhu intent on wakefulness? Here, during the day, while walking back and forth and sitting, a bhikkhu purifies his mind of obstructive qualities. In the first watch of the night, while walking back and forth and sitting, he purifies his mind of obstructive qualities. In the middle watch of the night he lies down on the right side in the lion’s posture, with one foot overlapping the other, mindful and clearly comprehending, after noting in his mind the idea of rising. After rising, in the last watch of the night, while walking back and forth and sitting, he purifies his mind of obstructive qualities. It is in this way that a bhikkhu is intent on wakefulness. (e.g., AN 3.16)
Several suttas precede descriptions of mindfulness of breathing or descriptions of the four jhanas with the practitioner sitting down, folding the legs crosswise, and purifying the mind of the five hindrances (see, for example, AN 10.60, MN 118, MN 27, MN 140, AN 3.63, etc). This might seem to imply that sitting meditation is required for these practices.
Then again, there is one sutta, AN 3.63, in which the Buddha describes how he attains any of the four jhanas and then, “when [he] is in such a state”, he walks back and forth.
Thank you for these helpful replies. One of the challenges in learning Buddhism is separating the commentaries from the suttas. So often I see statements about what the Buddha said or recommended with no corresponding sutta listed, then on further inquiry it turns out that it is from the commentaries, not the suttas. Generally, I try to disregard practice recommendations which cannot point to a sutta or direct quote from the Buddha. However, in this case (i.e., walking meditation is superior for insight), I have noticed in my own personal practice that at times I do have insights while on walking meditation. The flip side of this is that I cannot understand how one can attain the samadhi that is usually a precursor to insight while on walking meditation (because if one is too immersed in samadhi, one will walk into a tree, for example). AN 3.63 which you mention is interesting because it suggests that the Buddha first enters into the jhanas and thus attains clarity of mind through samadhi, likely through sitting meditation (as recommended in the Anapanasati sutta), then may go into walking meditation, which may provide opportunities for insight as one interacts with the world (through walking). AN 3.63 states:
“I collect some grass or leaves that I find there into a pile and then sit down. Having folded my legs crosswise and straightened my body, I establish mindfulness in front of me. Then, secluded from sensual pleasures, secluded from unwholesome states, I enter and dwell in the first jhāna…
“Then, brahmin, when I am in such a state, if I walk back and forth, on that occasion my walking back and forth is celestial/divine/noble. If I am standing, on that occasion my standing is celestial/divine/noble…” SuttaCentral
Addendum: I later came across an interesting Wikipedia post on this topic which seems to have credible references: Dhyāna in Buddhism - Wikipedia
See the post below for further clarification: Suttas related to walking meditation?
It might be helpful to remember that there are different ways of developing samādhi that supports insight, not all of which entail jhāna. AN 4.41 speaks to this, describing four ways of developing samādhi.
Bhikkhus, there are these four developments of concentration. What four? (1) There is a development of concentration that leads to dwelling happily in this very life. (2) There is a development of concentration that leads to obtaining knowledge and vision. (3) There is a development of concentration that leads to mindfulness and clear comprehension. (4) There is a development of concentration that leads to the destruction of the taints.
(1) "And what, bhikkhus, is the development of concentration that leads to dwelling happily in this very life? Here, secluded from sensual pleasures, secluded from unwholesome states, a bhikkhu enters and dwells in the first jhana, which consists of rapture and pleasure b om of seclusion, accompanied by thought and examination. With the subsiding of thought and examination, he enters and dwells in the second jhana, which has internal placidity and unification of mind and consists of rapture and pleasure born of concentration, without thought and examination. With the fading away as well of rapture, he dwells equanimous and, mindful and clearly comprehending, he experiences pleasure with the body; he enters and dwells in the third jhana of which the noble ones declare: "He is equanimous, mindful, one who dwells happily.’ With the abandoning of pleasure and pain, and with the previous passing away of joy and dejection, he enters and dwells in the fourth jhana, neither painful nor pleasant, which has purification of mindfulness by equanimity. This is called the development of concentration that leads to dwelling happily in this very life.
(2) "And what is the development of concentration that leads to obtaining knowledge and vision? Here, a bhikkhu attends to the perception of light; he focuses on the perception of day thus: 'As by day, so at night; as at night, so by day.’ Thus, with a mind that is open and uncovered, he develops a mind imbued with luminosity. This is the development of concentration that leads to obtaining knowledge and vision.
(3) "And what is the development of concentration that leads to mindfulness and clear comprehension? Here, a bhikkhu knows feelings as they arise, as they remain present, as they disappear; he knows perceptions as they arise, as they remain present, as they disappear; he knows thoughts as they arise, as they remain present, as they disappear. This is the development of concentration that leads to mindfulness and clear comprehension.
(4) "And what is the development of concentration that leads to the destruction of the taints? Here, a bhikkhu dwells contemplating arising and vanishing in the five aggregates subject to clinging: ‘Such is form, such its origin, such its passing away; such is feeling … such is perception … such are volitional activities … such is consciousness, such its origin, such its passing away.’ This is the development of concentration that leads to the destruction of the taints.
IMHO, this may be referring to first entering into Jhana, exiting into Samma Samadhi of mind and then doing walking meditation. 
AFAIK, its not possible to walk during Jhana, (leaving aside the disputed ‘insight jhanas’) but it is possible with the mind in Samma Samadhi. And AN4.94 seems to suggest that it is possible to develop varying levels of insight/ samadhi based on kind of practice and personal predilection.
Perhaps one of the venerables could shed some light on this?
There is an interesting related thread on this topic in Discourse here: When the Buddha is not in Jhana, what state is he in? - #9 by faujidoc1
I was re-reading AN 3.63 At Venāgapura (Venāgapurasutta) available here at SuttaCentral (Christopher had mentioned this sutta in his earlier post on this thread, and I just carefully reread it)
The key passage in regards to jhanas and walking meditation is the phrase:
Then, brahmin, when I am in such a state, if I walk back and forth, on that occasion my walking back and forth is celestial
The use of the word “such a state” (in Bhikkhu Bodhi’s translation; in Bhikkhu Sujato’s translation, he uses “this state”) implies that the jhana is continuing while walking.
This raises an interesting question regarding the jhanas: is it a state of such deep absorption that one is unaware of the world (which is what I originally mentioned in an earlier post on this thread, I think), or is interacting with the world (and thus able to walk while in a jhana state)? The phrasing of this sutta implies that one can interact with the world even when in a jhana state.
From my limited knowledge, I believe the notion that the jhana states are ones in which one is unaware of the world, derive mainly from the commentaries, not the early suttas. There is an interesting Wikipedia post on the topic https://en.wikipedia.org/wiki/Dhyāna_in_Buddhism:
Alexander Wynne states that the dhyāna-scheme is poorly understood.[50] According to Wynne, words expressing the inculcation of awareness, such as sati, sampajāno, and upekkhā, are mistranslated or understood as particular factors of meditative states,[50] whereas they refer to a particular way of perceiving the sense objects:[50]
Thus the expression sato sampajāno in the third jhāna must denote a state of awareness different from the meditative absorption of the second jhāna (cetaso ekodibhāva). It suggests that the subject is doing something different from remaining in a meditative state, i.e. that he has come out of his absorption and is now once again aware of objects. The same is true of the word upek(k)hā: it does not denote an abstract ‘equanimity’, [but] it means to be aware of something and indifferent to it […] The third and fourth jhāna-s, as it seems to me, describe the process of directing states of meditative absorption towards the mindful awareness of objects.[51]
Ṭhānissaro Bhikkhu, a western teacher in the Thai Forest Tradition, argues that the Visuddhimagga deviates from the Pāḷi Canon in its description of the jhānas, and warns against the development of strong states of concentration.[52] Arbel describes the fourth jhāna as “non-reactive and lucid awareness,” not as a state of deep concentration.[8]
In this framework, it is conceivable that a practitioner could be in the fourth jhana, then stand and walk or do other activities, all while remaining in the fourth jhana. This has profound implications for our daily practice and differs from what most contemporary jhana teachers present.
Please note that this sutta has no parallel. While this doesn’t mean that it’s not genuine, it should cause us to be cautious in relying on it. To my knowledge, there is no other sutta that indicates that one can walk during jhana.
Sure. I must warn, however, that these are subtle points. It is no easy matter to have a truly informed opinion on this.
Let’s start with the passage in question from AN 3.63. Ven. Bodhi does indeed render this as “when I am in such a state”, which seems to imply that the walking is happening while one is in jhāna. The problem, as so often, is that translation is an imprecise business. To be clear about the actual implication, we really need to go back to the Pali. We also need to cross-reference to other passages that have a bearing on this matter. But first of all, let’s see how Bhante Sujato renders the same phrase:
When I’m practicing like this, if I walk meditation, at that time I walk like the gods.
So ce ahaṃ, brāhmaṇa, evaṃbhūto caṅkamāmi, dibbo me eso tasmiṃ samaye caṅkamo hoti.
Here there is a significant shift in meaning. Whereas Bhikkhu Bodhi’s translation suggests the two are happening simultaneously, Bhante Sujato’s rendering is more open-ended. It suggests that during the general period one is practising the jhānas, the experience of walking, etc., will be divine. In other words, the point may be that the jhāna and the walking happen in close proximity to each other, not simultaneously. So which interpretation is right?
The critical Pali word here is evaṃbhūto, evaṃ + bhūto. The meaning of evaṃ is straightforward. Any dictionary will tell you that it means “thus”, “in this way”, etc., and that it normally refers back to something expressed earlier. Bhūto is the past participle of the bhavati, “to be”, and as such it means “been”. So the literal meaning is “been thus”. Now this does suggest something that happened in the past and that is no longer the case, bolstering Bhante Sujato’s rendering. But again we need to be careful. The broad usage of the flexible verb “to be” - and this as true for Pali as it is for English - means the range of idiomatic usage is significant. Bhūto often has the meaning “become”, in which case we get “become thus” for evaṃbhūto, which is perhaps closer to Bhikkhu Bodhi’s translation. We are not left with an unambiguous meaning.
Let’s go a bit further afield. At MN 122 we find a parallel construction where the meaning is far clearer:
It’s when a mendicant, quite secluded from sensual pleasures, secluded from unskillful qualities, enters and remains in the first absorption … second absorption … third absorption … fourth absorption … They focus on the imperturbable … While a mendicant is practicing such meditation, if their mind inclines to walking, they walk … While a mendicant is practicing such meditation, if their mind inclines to talking … While a mendicant is practicing such meditation, if their mind inclines to thinking …
Idhānanda, bhikkhu vivicceva kāmehi vivicca akusalehi dhammehi … paṭhamaṃ jhānaṃ upasampajja viharati … dutiyaṃ jhānaṃ … tatiyaṃ jhānaṃ catutthaṃ jhānaṃ … upasampajja viharati so āneñjaṃ manasi karoti … Tassa ce, ānanda, bhikkhuno iminā vihārena viharato caṅkamāya cittaṃ namati, so caṅkamati … Tassa ce, ānanda, bhikkhuno iminā vihārena viharato kathāya cittaṃ namati … Tassa ce, ānanda, bhikkhuno iminā vihārena viharato vitakkāya cittaṃ namati …
Here the situation is very similar to what we find at AN 3.63, except the meditation is even deeper. After the four jhānas, the meditator goes on to other meditations, including the immaterial attainments, here signified by “the imperturbable”. Again, the sutta describes a meditator who is walking “while practising” such meditation. But this time it goes further: not only is the meditator walking, they are even talking and thinking, vitakka. Now we know from the standard jhāna formulas that all vitakka ceases in the second jhāna. We know from other suttas that all speech ceases before jhāna. It follows from this that “while a mendicant is practicing such meditation” cannot mean that the meditation and the activity described is happening at the same time. It must be more elliptic, meaning something like “around the same time”, or “within the same time period”.
If this is true for MN 122, then it is also likely to be true for AN 3.63, where the exact meaning is even less clear.
The above shows how careful we need to be in interpreting the suttas. It takes broad knowledge and careful investigation. Sometimes relying on the commentaries is not a bad idea. We should be absolutely clear that a commentary is wrong before we reject its interpretation. The ancient monastics were learned and wise, usually more so than most modern commentators. It is all too easy to overestimate our own understanding at the expense of commentaries that have largely withstood the test of time.
And just to be clear, here is the commentarial interpretation of AN 3.63:
… samāpattito vuṭṭhāya caṅkamantassāpi caṅkamo dibbacaṅkamoyeva.
… the walking is a divine walking for one who is walking after arising from the attainment.
Thank you for your knowledge and careful analysis. There are so many things like this that really need an expert’s approach. I found it very useful how thoughtfully you laid your method out for us.
I’m not sure if you’ve read this thread but the question asked by the poster is similar in nature to yours. There is a response here from Bhante Sujato were he is relating a story told by a lay disciple who was on a meditation retreat.
https://discourse.suttacentral.net/t/an3-63-walking-in-4th-jhana/6633/5
Here is the snippet if you don’t want to read the whole post.
I think the Pali term for ‘walking meditation’ is caṅkamantanti 經行 (SN 14.15 = SA 447; see p. 142 in Choong Mun-keat Fundamental Teachings of Early Buddhism). But the term simply means ‘walking to and fro’.
AN8.63 is an important sutta for me and I’ve applied it to walking meditation. The sutta starts with a rather funny yet serious interchange with an unnamed (!) mendicant:
AN8.63:1.1: Then a mendicant went up to the Buddha, bowed, sat down to one side, and said to him,
AN8.63:1.2: “Sir, may the Buddha please teach me Dhamma in brief. When I’ve heard it, I’ll live alone, withdrawn, diligent, keen, and resolute.”
AN8.63:1.3: “This is exactly how some foolish people ask me for something. But when the teaching has been explained they think only of following me around.”
What’s interesting about this sutta is that the description of practice is not the usual four jhana list. Indeed it’s an interesting contrast:
AN8.63:1.6: “Well then, mendicant, you should train like this: ‘My mind will be steady and well settled internally. And bad, unskillful qualities that have arisen will not occupy my mind.’ That’s how you should train. When your mind is steady and well settled internally, and bad, unskillful qualities that have arisen don’t occupy your mind, then you should train like this: ‘I will develop the heart’s release by love. I’ll cultivate it, make it my vehicle and my basis, keep it up, consolidate it, and properly implement it.’ That’s how you should train.
The refrain that follows is repeated throughout the sutta:
When this immersion is well developed and cultivated in this way, you should develop it while placing the mind and keeping it connected. You should develop it without placing the mind, but just keeping it connected. You should develop it without placing the mind or keeping it connected. You should develop it with rapture. You should develop it without rapture. You should develop it with pleasure. You should develop it with equanimity.
In other words, for each meditation focus, this sequence of instructions repeats.
I found this perspective startling and useful from the very first time I read it. Indeed, for every activity in my life, I recall this instruction. And it’s been remarkably effective. Even for something as mundane as cooking, I am “placing the mind and keeping it connected, etc.” And the same holds true for walking meditation listening to the suttas.
The sutta does mention walking:
AN8.63:10.1: When this immersion is well developed and cultivated in this way, wherever you walk, you’ll walk comfortably. Wherever you stand, you’ll stand comfortably. Wherever you sit, you’ll sit comfortably. Wherever you lie down, you’ll lie down comfortably.”
My mnemonic for this sutta is “following me around”. I find it oddly…comforting.
Thank you so much for pointing to this sutta!
This passage reminds me very much of what Ajahn Maha Chatchai describes in his way to teach metta meditation:
AN8.63:1.6: “Well then, mendicant, you should train like this:
‘My mind will be steady and well settled internally. And bad, unskillful qualities that have arisen will not occupy my mind.’
That’s how you should train.
When your mind is steady and well settled internally, and bad, unskillful qualities that have arisen don’t occupy your mind, then you should train like this:
‘I will develop the heart’s release by love. I’ll cultivate it, make it my vehicle and my basis, keep it up, consolidate it, and properly implement it.’
That’s how you should train.
And this metta meditation has always been of utmost importance to me since I first listened to a recording of a course on it taught by Bhante @sujato. On this occasion also my heartfelt thanks to Bhante!
This meditation method starts out by, yes, “making the mind steady and well settled internally”, and then proceeds to the develpment of the actual metta meditation. 
Interesting parallel between the two suttas, in that they both say that when practicing a way of deep meditation, in whichever physical posture one is in, one is comfortable. AN 3.63 just replacing the word “comfortable” by “like the gods”, “like Brahma”, or “like the Noble Ones”.
I can see how AN8.63 can be an important sutta because it does provide a nice roadmap for an initial multi-layered meditation practice. In contrast, the Anapanasati sutta (MN118) has a very useful step-by-step approach for mindfulness of breathing, but it doesn’t include the pre-meditation steps described in AN8.63, and it does not include other important practices, such as metta.
Initially, when starting my study of how meditation was described in the suttas, I turned to the Satipatthana sutta (MN10) because it seemed so comprehensive, but I later learned that it may not be an EBT text and perhaps is a later compilation. What is also interesting about MN10 is that I think it omits metta meditation (please correct me if I am wrong).
Thus, AN8.63 could potentially be viewed as another useful initial roadmap for diverse meditation approaches because it does include the metta practices as well. I think it is missing the detailed description of the jhanas, although others, such as frankk, have mentioned that AN8.63 may be describing the first jhana when it mentions vitakka and vicara.
I am not sure whether it is to be considered “initial” if a practice guides you all the way to awakening …? 
And, yes, I’d say all the factors of at least the first and second Jhanas are present in the description in what Karl calls the “refrain”. The phrase “without rapture” even alludes to the third Jhana, and “with equanimity” to the fourth where this quality becomes most prominent. To me it doesn’t look so “initial”—nor, as it seems, to that mendicant, since they end up as an Arahant.
Or maybe it’s just my excitement over being pointed to a Sutta that I love. 
Ayya, I hope you 've had a chance, or will have a chance, to go to Wat Phleng Bang Phlat (near BKK) to be a part of a Sunday morning Metta meditation and talk with Ajahn Maha Chatchai. I have gone a few times, and though I don’t speak very much Thai, the atmosphere in the room is just so calming, so sabai, that I hope you get a chance to visit some day. I’m so grateful to Bhante Sujato for first mentioning him years ago, and his Metta teachings…it’s been such an important part of my practice in recent years.
How wonderful, Michael! Thank you for sharing! I can relate all too well to what you say!
You raise a good point. I had noticed the mention of rapture fading and equanimity arising (which as you point out, is the fourth jhana), but I guess I was looking for the standard description of the jhanas that occur in the suttas and thus initially missed it. But, this discourse is meant to be “A teaching in brief” so perhaps that is why the Buddha elides some of the standard jhana terminology.
This sutta and its surprisingly few parallels deserve much greater attention.
