Point of conceptual clarification. As far as I can tell, income mobility studies don't actually study mobility in the sense of the ability to move. They study actual movement in incomes. Mobility is a dispositional term. If I have been immobilized, I am prevented from moving. If I am immobile, but not immobilized, then I could have moved, but didn't. If I sit in my chair all day, my measured physical movement for the day will be low, but I may also be a spectacularly mobile person, able to run marathons, climb sheer rock faces, swim channels, etc. What we are interested in normatively from economic measures of mobility is whether there are structural barriers to upward movement, especially for the less wealthy, not the average deviation from parents' earnings. Can people earn more if they try? Once the average income reaches a certain threshold of material comfort, we should expect people's labor market choices to reflect preferences for many things other than income. So relatively low measured mobility (generation 2's income highly correlated to generation 1's) could indicate that people are fairly well satisfied with their parents' level of income and are optimizing on other margins. The better off people become materially, the less you ought to expect actual measured intergenerational movement in average income to reliably indicate the opportunity to move.
I have an extraordinarily interesting job that probably pays about 1/3 of what I could get on the labor market doing (for me) much more boring things, and so here I am happily foregoing twice what I actually make not to be bored. (So: what are my really real wages?) It turns out that this choice keeps my income in the neighborhood of my Dad's at my age, I'm guessing. (I, however, don't have a wife and three kids to support!) I am incredibly grateful to be at liberty, both economically and socio-culturally, to make this kind of tradeoff between income and satisfaction. And I'm sure I'm not alone.
The opportunity to make this kind of tradeoff in the labor market is largely a function of education. I think our current system of public schooling does create a structural barrier to upward movement for many of the least well-off, which is why we should scrap that system and replace it with a market in education as a matter of justice. But that's a matter of particular barriers to upward movement, which are what we should focus on, not some meaningless-by-itself average.