I know at this time, the people in the Tarim Basin were crowded around the glacier fed streams and when the glaciers completely melted away, the rivers dried up and they moved too.
The Hakka made five migrations across China, north to south. Many had Caucasian features when they left and were slaughtered during these migrations across China. To this day, they are known as 'the guests.'
My local Chinese restaurant owner was shocked when I said, "you're Hakka, aren't you?" He was.
Modern Japanese (with a heavy Hakka component) do have noses with high bridges. This occurs in families with individuals with smaller noses.
I have not yet figured out the name "Hakka" so I can't tell if it's their own name for themselves, or the Chinese name for them.