Noah Kristula-Green, based on data from Pew, showing how likely Americans born in each quintile of income are to end up in each quintile. As you can see, there is quite a lot of mobility, except that 41% of those born at the bottom end up at the bottom and 41% of those born at the top end up at the top.
On a more personal level, I see that children born into the 4th quintile have an 11 percent chance of ending up at the bottom, where some of my own offspring seem to be headed. Entirely by their own choice, I might note. One thing missing from most discussions of income mobility in America is this element of choice. I could easily have ended up in the top quintile had I really wanted to, but I didn't, and there are plenty of middle class kids who slide down the ladder because they refuse to pay the price of middle class membership.