Let me reproduce in the following Sri Aurobindo’s letters on Avatarhood. (Letters on Yoga, pp. 401-08)

Avatarhood would have little meaning if it were not connected with the evolution. The Hindu procession of the ten Avatars is itself, as it were, a parable of evolution. First the Fish Avatar, then the amphibious animal between land and water, then the land animal, then the Man-Lion Avatar, bridging man and animal, then man as dwarf, small and undeveloped and physical but containing in himself the godhead and taking possession of existence, then the rajasic, sattwic, nirguna Avatars, leading the human development from the vital rajasic to the sattwic mental man and again the overmental superman. Krishna, Buddha and Kalki depict the last three stages, the stages of the spiritual development—Krishna opens the possibility of overmind, Buddha tries to shoot beyond to the supreme liberation but that liberation is still negative, not returning upon earth to complete positively the evolution; Kalki is to correct this by bringing the Kingdom of the Divine upon earth, destroying the opposing Asura forces. The progression is striking and unmistakable.

As for the lives in between the Avatar lives, it must be remembered that Krishna speaks of many lives in the past, not only a few supreme ones, and secondly that while he speaks of himself as the Divine, in one passage he describes himself as a Vibhuti, vrsnīnām vāsudevah. We may therefore fairly assume that in many lives he manifested as the Vibhuti veiling the fuller Divine Consciousness. If we admit that the object of Avatarhood is to lead the evolution, this is quite reasonable, the Divine appearing as Avatar in the great transitional stages and as Vibhutis to aid the lesser transitions.

It [the overmind liberation] can't be supreme if there is something beyond it—but there is a liberation even in higher Mind. But in speaking of supreme liberation I was simply taking the Buddhist-Adwaita view for granted and correcting it by saying that this Nirvana view is too negative. Krishna opened the possibility of overmind with its two sides of realisation, static and dynamic. Buddha tried to shoot from mind to Nirvana in the Supreme, just as Shankara did in another way after him. Both agree in overleaping the other stages and trying to get at a nameless and featureless Absolute. Krishna on the other hand was leading by the normal course of evolution. The next normal step is not a featureless Absolute, but the supermind. I consider that in trying to overshoot, Buddha like Shankara made a mistake, calling away the dynamic side of the liberation. Therefore there has to be a correction by Kalki.

I was of course dealing with the ten Avatars as a “parable of the evolution”, and only explaining the interpretation we can put on it from that point of view. It was not my own view of the thing that I was giving.

Too much importance need not be attached to the details about Kalki – they are rather symbolic than an attempt to prophesy details of future history. What is expressed is something that has to come, but it is symbolically indicated, no more.
So too, too much weight need not be put on the exact figures about the Yugas in the Purana. Here again the Kala and the Yugas indicate successive periods in the cyclic wheel of evolution,—the perfect state, decline and disintegration of successive ages of humanity followed by a new birth—the mathematical calculations are not the important element. The argument of the end of the Kali Yuga already come or coming and a new Satya Yuga coming is a very familiar one and there have been many who have upheld it.

I only took the Puranic list of Avatars and interpreted it as a parable of evolution, so as to show that the idea of evolution is implicit behind the theory of Avatarhood. As to whether one accepts Buddha as an Avatar or prefers to put others in his place (in some lists Balaram replaces Buddha), is a matter of individual feeling. The Buddhist Jatakas are legends about the past incarnations of the Buddha, often with a teaching implied in them, and are not a part of the Hindu system. To the Buddhists Buddha was not an Avatar at all, he was the soul climbing up the ladder of spiritual evolution till it reached the final stage of emancipation—although Hindu influence did make Buddhism develop the idea of an eternal Buddha above, that was not a universal or fundamental Buddhistic idea. Whether the Divine in manifesting his Avatarhood could choose to follow the line of evolution from the lowest scale, manifesting on each scale as a Vibhuti is a question again to which the answer is not inevitably in the negative. If we accept the evolutionary idea, such a thing may have its place.

If Buddha taught something different from Krishna, that does not prevent his advent from being necessary in the spiritual evolution. The only question is whether the attempt to scale the heights of an absolute Nirvana through negation of cosmic existence was a necessary step or not, having a view to the fact that one can make the attempt to reach the Highest on the neti neti as well as the iti iti line.

He [Buddha] affirmed practically something unknowable that was Permanent and Unmanifested. Adwaita does the same. Buddha never said he was an Avatar of a Personal God but that he was the Buddha. It is the Hindus who made him an Avatar. If Buddha had looked upon himself as an Avatar at all, it would have been as an Avatar of the impersonal Truth.

I don't know that historically there could have been any other Buddha. It is the Vaishnava Puranas, I think, that settled the list of Avatars, for they are all Avatars of Vishnu according to the Purana. The final acceptance by all may have come later than Shankara, after the Buddhist-Brahminic controversy had ceased to be an actuality. For some time there was a tendency to substitute Balarama's name for Buddha's or to say that Buddha was an Avatar of Vishnu, but that he came to mislead the Asuras. He is evidently aimed at in the story of Mayamoha in the Vishnu Purana.

If a Divine Consciousness and Force descended and through the personality we call Buddha did a great work for the world, then Buddha can be called an Avatar—the tapasya and arriving at knowledge are only an incident of the manifestation.

If on the other hand Buddha was only a human being like many others who arrived at some knowledge and preached it, then he was not an Avatar – for of that kind there have been thousands and they cannot be all Avatars.

Krishna is not the supramental Light. The descent of Krishna would mean the descent of the overmind Godhead preparing, though not itself actually, the descent of supermind and Ananda. Krishna is the Anandamaya; he supports the evolution through the overmind leading it towards the Ananda.

One can be the head of a spiritual organisation or the Messiah of a religion or an Avatar without in this life reaching the supermind and beyond.

Yuge yuge [about his many births Krishna says in the Gita, sambhavāmi yuge yuge. See Gita, Ch. IV, 8] may be used in a general sense, as in English “from age to age” and not refer technically to the yuga proper according to the Puranic computation. But the bahūni [bahūni me vyatitāni janmāni tava cārjuna. Gita, Ch. IV, 5] has an air of referring to very numerous lives especially when coupled with tava ca. In that case all these many births could not be full incarnations,—many may have been merely Vibhuti births carrying on the thread from incarnation to incarnation. About Arjuna's accompanying him in each and every birth, nothing is said, but it would not be likely—many, of course.

But each being in a new birth prepares a new mind, life and body—otherwise John Smith would always be John Smith and would have no chance of being Piyusha Kanti Ghose. Of course inside there are old personalities contributing to the new life—but I am speaking of the new visible personality, the outer man, mental, vital, physical. It is the psychic being that keeps the link from birth to birth and makes all the manifestations of the same person. It is therefore to be expected that the Avatar should take on a new personality each time, a personality suited for the new times, work, surroundings. In my own view of things, however, the new personality has a series of Avatar births behind him, births in which the intermediate evolution has been followed and assisted from age to age.

I suppose very few recognised him [Krishna] as an Avatar, —certainly it was not at all a general recognition. Among the few those nearest him do not seem to have counted—it was less prominent people like Vidura etc.

Those who were with Krishna were in all appearance men like other men. They spoke and acted with each other as men with men and were not thought of by those around them as gods. Krishna himself was known by most as a man—only a few worshipped him as the Divine.

An Avatar, roughly speaking, is one who is conscious of the presence and power of the Divine born in him or descended into him and governing from within his will and life and action; he feels identified inwardly with this divine power and presence.

A Vibhuti is supposed to embody some power of the Divine and is enabled by it to act with great force in the world, but that is all that is necessary to make him a Vibhuti: the power may be very great, but the consciousness is not that of an inborn or indwelling Divinity. This is the distinction we can gather from the Gita which is the main authority on this subject. If we follow this distinction, we can confidently say from what is related of them that Rama and Krishna can be accepted as Avatars; Buddha figures as such although with a more impersonal consciousness of the Power within him. Ramakrishna voiced the same consciousness when he spoke of Him who was Rama and who was Krishna being within him. But Chaitanya's case is peculiar; for according to the accounts he ordinarily felt and declared himself a bhakta of Krishna and nothing more, but in great moments he manifested Krishna, grew luminous in mind and body and was Krishna himself and spoke and acted as the Lord. His contemporaries saw in him an Avatar of Krishna, a manifestation of the Divine Love.

Shankara and Vivekananda were certainly Vibhutis; they cannot be reckoned as more, though as Vibhutis they were very great.

It was not my intention to question in any degree Chaitanya's position as an Avatar of Krishna and the Divine Love. That character of the manifestation appears very clearly from all the accounts about him and even, if what is related about the appearance of Krishna in him from time to time is accepted, these outbursts of the splendour of the Divine Being are among the most remarkable in the story of the Avatar. As for Sri Ramakrishna, the manifestation in him was not so intense but more many-sided and fortunately there can be no doubt about the authenticity of the details of his talk and action since they have been recorded from day to day by so competent an observer as Mahendranath Gupta. I would not care to enter into any comparison as between these two great spiritual personalities: both exercised an extraordinary influence and did something supreme in their own sphere.

He [Ramakrishna] never wrote an autobiography—what he said was in conversation with his disciples and others. He was certainly quite as much an Avatar as Christ or Chaitanya.

Mahomed would himself have rejected the idea of being an Avatar, so we have to regard him only as the prophet, the instrument, the Vibhuti. Christ realised himself as the Son who is one with the Father—he must therefore be an amśāvatāra, a partial incarnation.

What Leonardo da Vinci held in himself was all the new age of Europe on its many sides. But there was no question of Avatarhood or consciousness of a descent or pressure of spiritual forces. Mysticism was no part of what he had to manifest.

Avatarhood: Letters on Yoga, pp. 414-15:
I accept it for Rama because he fills a place in the scheme—and seems to me to fill it rightly—and because when I read the Ramayana I feel a great afflatus which I recognise and which makes of its story—mere faery-tale though it seems—a parable of a great critical transitional event that happened in the terrestrial evolution and gives to the main character's personality and action a significance of the large typical cosmic kind which these actions would not have had if they had been done by another man in another scheme of events. The Avatar is not bound to do extraordinary actions, but he is bound to give his acts or his work or what he is—any of these or all—a significance and an effective power that are part of something essential to be done in the history of the earth and its races. All the same, if anybody does not see as I do and wants to eject Rama from his place, I have no objection—I have no particular partiality for Rama—provided somebody is put in who can worthily fill up the gap his absence leaves. There was somebody there, Valmiki's Rama or another Rama or somebody not Rama.