The Economist Greg Clark, who you may have heard of as the author of A Farewell to Alms, had another book, The Son Also Rises, about social mobility - this time showing that it is quite slow in both egalitarian societies like Sweden and inegalitarian ones like Chile. I have heard reference to "the surname paper" a few times over the years but had never followed up on it. There is a New Republic interview with him, and there are a a few versions of the paper, of increasing length. This one is good.
Most cultures have some version of a proverb like "clogs to clogs in three generations." Three seems a little quick to those who think in terms of Old Money and land ownership, but it also sounds plausible. Clark says status, when measured by a variety of measures rather than a single one (as economists and other social scientists often do), actually last 10-15 generations. Even a lot of liberal outlets covered this when it came out. Even I can see that some of the criticism of it, some by professionals, is bogus. But I can't attest to other criticisms.
Much of his work is focused on England from 1200-1910 or so. Part of his contention is that more children of the rich survived, and thus are more common now. Hence the surnames as a proxy for that.