The study refers to people born in 1980-82 and "when they are adults" is age 30. Of course, 30-year-olds generally earn less than middle-aged people, but the authors of the study say that relative positions have pretty much stabilized by then--that is, we'd see about the same pattern if we came back 20 years later.
Here is the pattern for people whose parents were in the 60th percentile.
The large number of people in the second percentile (actually about 6% of all 30-year olds) had zero income. That makes it hard to read, so here is the figure showing just the lower part of the y-axis.
The most common destination is in the low 70s. The chances of rising above that level drop off pretty sharply. But overall, the differences are pretty small: you could say that people from the 60th percentile are about equally likely to end up at any point in the distribution.
Here is the 80th percentile. It's a similar basic pattern, although the chances of winding up near the top are higher and the chances of ending near the bottom are lower.
Here is people whose parents were in the 98th percentile. This looks different--the higher the ranking, the better your chances of getting there. The most likely destination is the 99th percentile--even higher than the "percentile" of zero earnings.