The Harrapa Ancestry Project does a decent job of covering this from a genetic standpoint. Dienekes also covers the subject well. I think the consensus is that there were migrations from West Asia into India that left a mark on the genetics of India, but that overall this contribution was minor, and that populations in Northern India were already a bit closer to populations in the Middle East and West Asia. So some of these divisions were pre-existing. Vedic Sankrit has loanwords that are believed to be Harappan in origin and elements of their culture may have survived in modern religious practices too.

Razib Khan did a great job summarizing David Reich's seminal paper on the subject. I highly recommend both if you're interested.

The main theory I ascribe to is that the Aryan migration did happen. The Aryans were a Indo-Iranic people, descended from the original Indo-Europeans. They began to move south from the central Asian steppes into the Punjab and Pakistan around 1700 BC, as members of the Gandhara Grave culture: http://www.royalasiaticsociety.org/site/files/Part%203%20-%20Aryans%20and%20Nomads.pdf They used passes in the Hindu Kush to move slowly and steadily southwards as part of a general trend of Indo-European expansion south, bringing horses, new weaponry, and new ideas. At the same time the Aryans were entering Northern India in mass, the Harappan civilization was collapsing.

There are some theories connecting the collapse of this civilization to some hypothetical Aryan invasion(argued mainly by the British archeologist Sir Mortimer Wheeler), but this theory has been largely disproven. There are not any good signs of violence in Harappan sites, and by the time the Aryans would have been a major presence in the Indus region the Harappans were already collapsing. The main theory nowadays is that a large drought completely destroyed the Harappan civilization, and caused it to degenerate into a less developed society called the Ochre colored pottery culture.

This culture was the main one present when the Aryans were heavily colonizing the Indus, and this culture combined with other local cultures and the Rigvedic Aryan civilization produced what we know as Vedic civilization. The main evidence of this merging is both archeological, in the form of new artwork and remains, and literary. The Rigveda(the oldest Veda that's dates back to the original Vedic migrations)mentions both the un-cremated forefathers, or ánagnidagdha(the pre-Aryan Harappas) and the new Aryan cremated dead, or agnidagdhá. This indicates a mixed Indian population at the time.

TL,DR: There was a Aryan migration but It most likely did not wholly cause the Harappan collapse.

Hope this is a good answer, I can't answer for the South Indians though.