Some would remain, but humans like animals radiate toward suitable environments, the places we lived 1,000 years ago may not be as suitable as they once were.
The deserts of SE California were covered with lakes and had a temperate climate 4,000 - 5,000 years ago. This changed and the most natives moved on, those remaining had to adopt different strategies.
OBTW: I don't think that this migration is a cognitive decision.
Much prior to that the Sahara desert was a populated grass land.
I think we need to give early man more credit. (Maybe if they were as stupid and lazy as some of us are now, we wouldn't exist.) I believe that early man migrated far more than what we think. In a mobile society, of coarse you would move if conditions became adverse. (And heck, no government to demand they fix it.)