You’ll see when I finish the writing 
(if I ever do…)
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Update:
I’ve found the Saṁyukta Āgama parallel version with translation. It is SA 374:
Personally I think this aids in the understanding that this is not talking about rebirth in this particular case. I definitely think avakkanti can and does refer to that quite often. But as we have seen, it certainly does not always mean it. This sutta also says that the nutriments “cause beings to abide in the world and receive nourishment,” different from the formulaic Pāḷi version (an interesting note). There are certainly other ways one could interpret it, but this one makes a lot of sense. Clearly future becoming, i.e. rebirth, is the result of volitions increasing. If the avakkanti of nāmarūpa refers to the same thing as punabbhava, it would already be “future becoming” before the growth of volitions and thus the punabbhava that would happen on account of that. If we take it as all happening at once, the growth of volitions is out of order and a bit nonsensical, and it has this order in both parallels in an easily explainable way.
Consciousness abiding in and increasing a particular “object” matches with “entering” into names and forms, and of course explains the subsequent “growth of volitions.” Like I said, one makes volitions, intentions, choices, generates kamma, etc. on the basis of particular stimuli (kamma is said to arise dependent on contact) and our relationship to them. This would of course establish one for future existence and rebirth when the time comes. I list some other suttas below where consciousness “abiding in” things is equated with it being supported on them with delight and lust. So the imagery of consciousness entering in, dwelling, etc. is supported for these things in the suttas. It also makes sense in the context of the usage of the word avakkanti elsewhere. Sometimes traditional understandings can blind-side an original rendering.
The other interesting thing is SA 377! I cannot find a solid English translation, but I was able to get a very rough one. It is a parallel that covers the simile of the painter in Pāḷi. However, this one makes reference to not being able to paint on a non-visible surface, much like the simile of painting on the sky in the Kakacūpama Sutta. Nāmarūpa cannot be ‘painted’ onto viññāṇa when there is no delight and lust in relation to the nutriments of it that it gets established in and grows with, causing for it not to “enter into”/abide there. Below I leave a rough translation with the Pāli.
This would connect to anidassana viññāṇa. which is said that the elements do not get a “footing” in it and it does not partake of the all-ness of the all (commonly misinterpreted by semi-eternalists as we have likely seen). This word (anidassana) is the same one used in the Kakacūpama Sutta, with the same simile and reference used in SN 12.64 / SA 377. Consciousness, without delight and lust, is like an anidassana surface that a painter cannot paint on. I.e. it does not get attached to what it cognizes, and as such, there is no rebirth and no more accumulation of dukkha according to the sutta. One could of course argue that this is separate from the anidassana viññāna of DN 11, but either way anidassana here is being used to explain viññāna that does not delight in or have greed / abide in what it cognizes. I suspected they could be connected in the Pali, and now seeing the Chinese parallel specifically mention it is interesting.
Of course consciousness of an arahant still cognizes things. But those things do not “paint” and color it with all kinds of impressions—it does not “enter into” the particular namarupas for the growth of sankharas, and as such, is not established and so it cannot be re-planted elsewhere and be reborn. The anidassana viññāna passages of course fit into this. Rather than referring to a separate type of consciousness, the adjectives describing the consciousness as something that is anidassana (cannot be drawn on, coloured, show up/be visible) and sabbato paham (“entirely given up/relinquished,” the more authentic recension than ‘luminous’ variants, though this is not 100% clear in the participle) point to it simply referring to plain old consciousness, but of the arahant. Other references to this same type of thing in the canon make clear it is not a mystical undying consciousness, but just referring to Nibbāna and eventually the extinguishment of all phenomena.
Wisdom ‘blocks out’ the streams of the elements finding a footing (I see this especially in terms of the above passage and in light of MN 1 where the elements represent the things of the world that we conceive, appropriate, delight in, etc.); and the final cessation of consciousness is the cessation of all nāmarūpa. There are other passages to support this but I just wanted to show this passage with some ideas. Briefly, Snp 4.11 connects the “disappearance” of form to a form of perception without papañcasaññāsankhā / conceit (and that as far as the structure of the sutta goes, clearly is building to an explanation of awakening).
Snp 4.15 is also very similar:
As for the previous paragraphs, there is frequent mentions in the discourses to consciousness “abiding in” or “dwelling in” or “getting stuck” in things referring to attachment and desire in relation to particular objects. The exact same language used here and throughout this general section of the nidāna samyutta. It not being established and growing / not entering into nāmarūpas and then forming sankhārās all match these references, and the cessation of them, equated to the painter who cannot paint on an invisible/formless surface, would simply be a consciousness not established upon anything in the same way described by Mahā Kaccāna and so forth.
Some examples of these references to consciousness “abiding” / dwelling in things—the same language used by the Āgama parallels here for the establishment of consciousness on a object via lust and delight and entering into nāmarūpas—are in MN 138 and SN 22.3. There are plenty more but those come to mind as I’m a fan of Mahā Kaccāna 
Note: This is assuming that anidassana in, say, MN 49 is not simply a copy from DN 11, as some scholars (such as Bhikkhu Anālayo) have suggested could be the case. Having been likely spoken by Baka Brahmā originally to refer to his attempt to go invisible and disappear, it may have been wrongly attributed to the Buddha (as in the commentaries) as somehow copied over or influenced by the passage now found in DN 11 (a similar sutta on the Buddha’s superiority to Brahmā). However, what is clear is that what is important is the cessation of consciousness and the non-established consciousness during the life of the arahant, referring to the fact that it does not get stuck or abide in external objects and simply is aware of them with wisdom until no longer arising. In fact, scholarship has shown that the original word was most likely pa(ja)ham, referring to consciousness having been “relinquished” and therefore leading to its eventual extinguishment. As such, this interpretation would fit with the suttas descriptions (which clearly align with MN 1, non-identification, non-greed/lust, etc.) and account for an interpretation that does not contradict any actual doctrinal messages.
If anyone reads classical chinese and can translate the āgamas, please let me know! It’s a short sutta:
Here is a horrible translation I’ve pieced together with a dictionary and translators/edits:
Much mettā!
Oh no! another fascinating thread that I now have to find time to read carefully! Thanks so much @Vaddha and @Sunyo and @knotty36 for this opportunity 
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I don’t want to argue any further, but since others are now reading this thread too, I would like to clarify one point where I may have oversimplified a bit for sake of argument. Namely, the mutual dependency of nāmarūpa and viññāṇa. I wish to clarify it is not purely about rebirth. I focused on that aspect here, since the topic is on avakkanti, which, as I explained, I do think is purely about rebirth. But of course the mutual dependency of consciousness and nāmarūpa continues after birth as well, and stands until the enlightened one passes away. Still, central to the idea is rebirth, which is obvious from the passage in the womb and others I quoted. The mutual dependency is about what kind of sentient existence there can be, basically. There is on the one hand no existence of pure consciousness without other factors, and on the other hand no existence without consciousness.
And existence starts at birth, that’s why rebirth is central.
I hope that makes sense. Perhaps there were some other slight simplifications on my side as well, since, again, I was mainly concerned with the topic at hand, namely avakkanti. We got dragged around a bit but I tried to sort of stay on that topic without mentioning all little nuances. If any misunderstandings arise because of this, I apologize.
Readers may also be interested in this essay, which is also on the topic of nāmarūpa and viññāṇa: Viññāṇa anidassana: the state of boundless consciousness. Here the dependency of the factors is sometimes ignored by certain interpretations who say there is a consciousness that stays after nāmarūpa ceases. I suggest an alternative interpretation of the passages in question, a reading which actually reinforces the dependency.
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I’m popping in at the end here to add a few things to previous comments.
I don’t have the time to review all of what we discussed here. However, I do want to clarify that I understand avakkanti as referring to rebirth in the context of nāmarūpa/viññāna. We looked at examples where it could potentially be read other ways or curiosities in the text, but still I do not find those readings conclusive in any way, and think avakkanti simply refers to the establishment of consciousness in a new nāmarūpa at conception. This way others reading above posts will not be misled by the speculations here
Mettā