Here is a writeup of using CRIU on openvpn: https://www.redhat.com/en/blog/using-criu-upgrade-vpn-server...

So it seems to work. But I'm no expert and I've never done it, so I might well be missing something.

Fairly sure files, network connections, ... are not a problem, unless I'm confusing it with a different migration solution. GUI apps probably only work if access through some network connection.