If you need to remove something from the Inbox, I would expect to find a NotesDocument.RemoveFromFolder() call.
In the target folder -- has it been changed to show documents in hierarchy, e.g.? Because if it has, it could get tangled up when you run NotesDocument.PutInFolder() on a response doc (that is, a doc with a $Ref field).
And what's your default folder design? If it changed to a hierarchical view, that'd account for a lot.
I'd expect the partial-hierarchies switch might also have some heartburn over moving only responses to the folder, but I'm not sure exactly how it's impacted.