"Air Force One", thank you, it may very well be related. I'm trying a matrix of tests to see where it fails and where it works. It seems that my initial assessment that this is a 6.0.4 regression which was working on 6.0.2 might be wrong. It seems to work initially, but after a couple of minutes it starts failing at the "Preparing to copy..." stage.
I hope that today I'll have a more complete picture with various guests and different GAs. Because it turns out that it's the GAs after all, not the main VirtualBox version. I've at least established that part...