These studies of intergenerational class mobility can be highly problematic, because they often just look at
income, very much less
access to opportunity. The reason being patterns like the following are not uncommon:
1. John is a banking executive who makes millions of dollars a year.
2. John has a son, Jack. Jack sees how much his father worked and decides to take a very different path, becoming a documentary filmmaker. From his family's wealth, Jack never worries much about money, but his nominal income is quite low.
3. Jack has a son Joe who decides he'd rather live in the giant estate of his grandfather than the small suburban house of his father, and goes into banking. He gets help from his grandfather's contacts to land his first job.
Thus, if you look at these 3 generations in isolation, you'd count them as going from the top quintile to the bottom quintile to the top quintile again. But, in reality, nothing about their social status has really changed, nor their access to opportunity.
There was a post on HN from a couple years ago that looked at a study that said the rich families from Florence, Italy in the 1400s were still the rich families today: https://news.ycombinator.com/item?id=18872376. The argument was that there was much less actual social mobility than some other studies suggest.