Here is the latest, by Emmanuel Saez and co-authors; note I linked to this paper yesterday but now I have looked at it. Here is one key sentence:
…we find that short-term and long-term mobility among all workers has been quite stable since the 1950s.
To disaggregate, note that mobility among males is down but mobility among females is up. (It is an interesting question whether there is a causal relationship here.)
Here is a much earlier MR post on mobility. Keep these links in mind next time you hear claims about mobility, and I believe you will hear many such claims in September. See also our earlier posts on Dalton Conley, who shows just how much inequality is generated within the same family.
It should be noted that Saez is the leading measurer of income inequality and also a critic of such inequality. In his view a constant level of mobility means that no force is offsetting ongoing inequality. I believe he would likely read his own paper as support for a left-wing view of the world and as support for concern with income inequality. He would not read his work as reason to dismiss the mobility issue. My view differs, as I worry about mobility — can a hard-working person get ahead? — but I do not worry about inequality per se, nor do I require of mobility that it overturn a particular level of inequality.