Here's a link from the LSE to get you started. Google is your friend: I searched on "comparing social mobility between europe and the US" and that popped up.
I suspect that maybe in Europe getting into the top 20% is harder, but in the US getting out of the bottom 20% is harder. Which way around is would be more desirable?
this is looking a lot bigger than I can take on right now, unfortunately. It's very interesting, but would require significant study to get it right.