There's an old documentary series from the UK where they sent kids from disadvantaged areas to the same schools as the wealthy. Took them all of a couple of months to pick it up and it would work in reverse too.

Yup. The physically, economically and socially mobile class is derived principally from people who moved from the working classes, mostly in the centuries following industrialisation.

The genetic divide goes more the other way - of course there’s going to be some positive selection for educational attainment/intelligence for people who left the village, but more generally the local populations are quite insular, migrate little (especially the ones who remain, of course - self-selecting), and have quite a few children, and in a population like that, you get genetic drift, resulting in more distinctive alleles compared to a generally larger mobile population, compared to the individual sedentary populations.

Where I live in Portugal this process has been going on for nearly a millennium - and you can tell if someone is from village X or Y 5km apart but separated by a river, just by looking at them - specific alleles get more and more prevalent in a small, largely closed population, quickly.

This doesn’t show that social mobility is broken - if anything, the opposite - it shows that a great many people have left the village and joined the mobile elite.