Best Funds for International Mobility

Currently Assoc at MF in London and keen to explore ways to move to the US in the next couple of years. Not a thing at my current fund, so I am trying to understand more generally:

1) Which funds have a track record of sending London/European people over?

2) When is the best time to make that move? (pre or post promotion)

3) If leaving at associate level, would you be able to be promoted without MBA? If so, at which funds (aware of Apollo, KKR, Silver Lake)?

Thanks!

Context: EU citizen, no US visa