Actually, that chart doesn't show who's in those groups over time at all.
People's views of their mobility are discussed extensively in chapter 4 (remember, this is largely the results of an opinion poll), but actual statistical mobility isn't really touched on (at least that I've been able to find. Long paper. If I missed it, flag me).
There is this in the overview:
In 2011, this middle-income tier included 51% of all adults; back in 1971, using the same income boundaries, it had included 61%. The hollowing of the middle has been accompanied by a dispersion of the population into the economic tiers both above and below.
The upper-income tier rose to 20% of adults in 2011, up from 14% in 1971; the lower-income tier rose to 29%, up from 25%.
If you do the math, the middle class dropped 10 points, the upper class rose 6 points and the lower class rose 4.
Then there's this (not from the paper):
The percentage numbers don't match, which I think is about where the different "splits" are made for income ranges. The chart shows its splits. Pew puts "middle income tier" at anywhere between $39,418 to $118,255 (footnote one in overview) which seems a bit generous since that latter number gets you comfortably into the top quintile.