Linguist here. The answer is "no" regarding a consensus, and if I were a betting man, I'd wager that of the scholars who study the AIT, the consensus is that it didn't happen.

I was taught the Aryan Invasion on linguistic grounds. Sanskrit and its daughter-languages are clearly Indo-European in origin, and in fact, Sanskrit was pivotal in the development of the theory of Proto-Indo-European. Dravidian languages, meanwhile, are all south of the I-E languages. Because of the modern north-south division, it's easy to postulate that the Aryans, known to have traveled as far west as Iceland and as far east as China, pushed the natives south (as Aryans are wont to do). The archeological and genetic evidence disagree greatly with the theory, though, which you can read about on Stephen Knapp's website. He includes the history of the theory.

For a different scholarly perspective, try Koenraad on the Aryan non-invasion theory. He covers much of the same evidence but instead seeks an answer as opposed to simply criticizing the other theory.

As for what we know about Aryans, the answer is "more than you'd think" but "not as much as we'd like." For example, there's no common word for "beach" in the I-E languages, so we know the homeland probably wasn't coastal. You can see the full lexicon thanks to the University of Texas. The homeland question is still unresolved though, with a 2012 archeological study and a 1992 genetic study both leaving many questions. Archeology has yielded some cool finds from burial sites. I remember reading about that in Heaven, Heroes, and Happiness by Winn, which I also remember it being a little dated, but still a good starting place. The evidence he cites doesn't change, just the theories.

One of the most interesting papers I read on the Aryans was a comparative mythology paper by Bruce Lincoln, "The Indo-European Myth of Creation." I have no idea how well received the article was, it's 30 years old at this point, but he postulates the origin myth of the Aryans was that there were twins, one a priest, who sacrifices the brother and a bull. From the bull came all animals, from the brother came more humans. Might be worth a read after "Heaven, Heroes, and Happiness."

Again, I'm merely a linguist, so if my answer doesn't quite hit the mark, I apologize.