This is a highly controversial subject, in fact those not familiar with the subject, will have no idea of the enormity of how controversial this is. It caused a huge stir when it was originally put forward by Max Muller and other European scholars, and modern Indologists have debunked the Aryan Invasion theory and in Indian nationalist politics the subject has become highly volatile and emotionalised.
I just wanted to preface my post with this.
Today, many indologists and especially Indian indologists dismiss Aryan Invasion theory(AIT) as racist and political propoganda by European scholars and Christian missionaries. That is because the theory has no evidence to support it(there are no records of any migrations) and its entire conception is based on invalid reasoning. Moreover, the recent evidence seems to suggest that there was no Aryan race and the the Rig Veda is indigenous to India based on archeological evidence, textual evidence and astronomical dating of the Vedas.
The word Arya itself is a Sanskrit word which means noble and there is no usage of this word in Vedic literature that refers to a race. The concept of an Aryan and Dravidian race was propounded by Max Muller based on physiological features of North Indians and South Indians. He noted that that the people in the south had darker skin and the people in the north had lighter skin and could only explain this by positing a racial dichotomy. This is fallacious by modern scientific standards and certainly modern genetic evidence does not show significant racial differences between North and South Indians. The difference in skin tone could be explained by taking into account others factors such as differences in climate.
The second evidence provided for AIT was a discovery by Sir Monier William Jones that Sanskrit belonged to the family of Indo-European languages, because the similarities were so remarkable, it would be impossible for it to be a coincidence. It was thus posited that all Indo-European languages belonged to Proto-Indo European(PIE) which was a lost language and has been reconstructed using linguistic science.
Thus strong evidence emerged that migrations had taken place between Indo-Europe, otherwise it is impossible to explain how Indo-Europe could have a common language family.
Then based on linguistics science to find the linguistic cente of gravity, they determined that the Aryans must have originated in Central Asia, in the mountains around Russia. They were a nomadic race that migrated around Indo-Europe and eventually arrived in India. This coupled with Max Mullers findings lead to the theory that a white-skinned Aryan race that spoke Vedic Sanskrit had invaded the dark-skinned Dravidian race and pushed them down South India. This also explained the sudden demise of the Indus Valley Civilisation(supposedly Dravidian)
Then Max Muller tried to produce evidence for his theory by trying to look for the mention of Aryan invasions in the Rig Veda. However, the translations rendered by Indian Pundits showed no such mentions. Their translations based on Sanskrit grammar and etymology, suggested a highly sophisticated people discussing metaphysics, monotheism and ethics which was also consistent with the Indian schools of Philosophy and Vedanta. This did not at all support Mullers theory of a nomadic and primitive people which spread through Indo-Europe through war using iron weapons and worshipped natural gods. So Muller and other European scholars accused the Indian Sanskrit Grammarians and Pundits of fabrication. They alleged that their grammatical and linguistic sciences were useless, and that they had more advanced methods of using comparative linguistics to translate the Vedas. Thus, declaring this, they rendered their own translations of the Vedas using their supposedly superior methods, with the aid of a commentary by a 10th century Indian commentator Sayana.
Muller did not remain loyal to Sayana's commentary either, and where he disagreed with a translation of a word, he would use his own. A problem was that Sayana provided an appendix of up to 30 different meanings for some words, so how could you know what a word actually meant? Some words were archaic, and Indian Pundits used the 'Nirukta' which was a supplemantary Indian text explaining the etymology of archaic words used in the Vedas, but Muller rejected it.
Somehow, using a hodge-podge of methods, Muller rendered a translation of the Rig Veda. It bared absolutely no resemblance to the translations the Indian Pundits had based on their Sanskrit Grammar and linguistic sciences. Mullers translation was more consistent with his theory of a nomadic and primitive people that worshiped nature gods and spread through war. On the other hand, the Indian translation suggested a highly philosophical people with scientific temper, that pondered over matters like the nature of reality, had elaborate cosmological and theological theories on monotheism and monoism and discussed a highly refined ethics.
Then after finishing his translation, Max Muller started to draw interpetations from them. Where it would read "Indra destroy the darkness and let light triumpth" Muller would interpret that the Aryans were racist who hated black skin(Dravidians) Similarily, where the Vedas would read "The Maruts are golden and shining and they aid Indra in dispelling the darkness" he would interpret that the Maruts are fair-skinned and were actual troops of a war-lord called Indra that ransacked the Dravidian cities.
In his time he was not the only one that came up with these interpretations. In fact many European scholars had a go at interpreting - some came up with themes of incest in the Vedas, some came up with homosexuality, and some with human sacrifice(the famous Purush Sukta in the Vedas describing how the supreme being fashioned the universe out of himself)
Today by modern scientific and academic standards the kind of scholarship Max Muller and other European scholars did would be clearly considered invalid. It is based on circular reasoning at best, and at worst, it is highly racist.
The nail in the coffin of Muller's scholarship was to date the 'Aryan invasion' around 1500 BCE using a biblical chronology where the world is created in 4004 BCE and the great flood happens sometime later.
In summary: AIT is a dead theory. There is no evidence of an Aryan race, Aryan invasions or Aryan migrations.