Up until about 4000 years ago ALL the Dravidian speaking people lived in the North. Then, with settlement possible in the South they moved there. Today we think of them all as living in Southern India.
Note, the big cats did great in the jungle in the South ~ and until the area dried out humans could not compete with them. China seems to have had some problem along that line up until the end of the Younger Dryas ~ then the climate changed, the cats languished, and people moved in. But up until that time people lived West of what is considered modern China.
I was a student of Chinese at the DLI in Monterey in the early 80's. At that time, the professors all had to have received at least a masters degree at a mainland China university. One of them had a masters in history.
He said that the tribal legends of the Han and minority tribes tell of a constant south-easterly migration, back to the earliest identifiable threads.
Yes, tribes. It seems that at least before Mao took over, most Chinese people knew which tribe and sub-grouping they were part of.
Your story is changing.
Please cite a scientific reference supporting your claim of significant climate change in India circa 2,000 BC.