When a thread is moved currently, a lock appears next to its listing. This causes me to believe the thread has been locked, not moved, and unless I feel compelled to, I don't usually go into the thread to investigate any further.
What I'm proposing is two-fold: 1) When a thread is moved, instead of showing the lock icon, show an arrow of some sort. 2) Inside of the thread, have an automated posting in the name of the moderator responsible stating the thread has been moved to x forum; also have a link directly to the new location. (The first request is a bit more important and seemingly easier to implement than the second.)
I have no idea whether this is at all simple to do. I do think it would help add clarity to the forums.