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.