Eh, no. Genetic stuides show that South Asians are descendants of two major ancestral components, one restricted to South Asia (Ancestral South Indian) and the other component (Ancestral North Indian) more closely related to those in Central Asia, West Asia and Europe.
Sure, the Aryan Invasion/Migration theory is too simplistic but nevertheless, there is a clear dicotomy.
N.B. Ancestral South Indian and Ancestral North Indian are testing used in genetic studies to sidestep the use of loaded/controversial terms like Aryan and Dravidian.
Disclaimer, I'll not a geneticist. And I've seen, read, or come across various studies supporting opposite viewpoints. Given that, it is non trivial to decide what is the ultimate truth and the truth we decide to pick depends on our own biases. And I don't think this question is settled and may never be settled.
My personal take, for what it's worth, there was no mass scale migration and North and South Indians are the same people. However, there was a lot on influx/migrations/invasions in North from rest of Asia and farther, which led to genetic mixing and the deviations that we see.