My understanding is that there is a very strong correlation between a person's educational and socio-economic attainment and those of their parents. The number that sticks in my head is that parental status accounts for 80%+.
If you are born to parents in the lowest quartile you have ~10% chance of making it into the highest by age 26. For people born in the highest the probability is ~30%.
(and if it was random it would, obviously, be 1/4 = 25%)
On its own, leaving a quartile doesn't say anything about the magnitude of the change, so it's not a very useful measure of anything.
Another problem is that because the distribution is so heavily skewed toward the top (for both income and wealth, but more heavily for wealth), dividing the population into equal groups creates a somewhat misleading picture of mobility. A nonlinear breakdown of classes is more reflective of the distribution. In that case we might ask something like how many people move from the bottom 40% to the top 5%.
Although for some questions, e.g. "Can you escape poverty in the US today?" (which could be specified as moving from 1st -> 3rd quartile) this data works.
But, for other types of questions, e.g. "Is extreme wealth mostly hereditary?" you'd need a non-linear breakdown of classes.