OK thanks so much for that tip, it led me right to the solution.
Yes you were right, the email WAS being moved.
IT turns out though that the folder she moves the emails into are a subfolder under the same email account, so the search is, correctly, still showing it.
So the fix is that she has to remember to change the search parameter from Current Mailbox, to Current Folder, then it does not show up after it has been moved.
Thanks