Hi Scott,
Add the Changed by field to the view - it *might* show the person's name who moved the messages. It will either show the name of the person who last did something to it (which isn't necessarily the person who moved it) or the shared mailbox name. It is easy to check and may give you enough information - and is available to any end-user.
The only other way is if you have audit logs enabled and are tracking this. Logging needs to be enabled before the folder is created or the messages moved and only the admin can review the logs.