Moving upmarket from growth firm?

Kind of depends on the types of deals you've been working on and how good your networking/hustle is. If you're at a decent name fund like a Susquehanna/Level/Lead Edge, that would be enough to get you into a conversation, but from there you'd have to convince them you could do the work, which will be the tough part.

In general it's tough, not just given size of the investment, but of the firms you've mentioned, most of those have become quasi buyout funds now. They still do some growth investments, but places like Thoma are just straight up doing majority deals. What that means is the skillset you'll need is much more technical at the Associate to VP level whereas I bet you've been more focused on sourcing, slightly less technical diligence, portco ops work etc which is all great experience, but it won't matter at a Silver Lake where they really just want you grinding on models and diligence stuff. Even AKKR and Great Hill do fewer minority deals now. If anything, I would be looking at the IVPs of the world if you're looking for large, later stage growth.