I was reminded that I was going to check something regarding the problem with people falling into/becoming part of the movers sometimes.
The only thing I could think of to cause this is a lack of a collision model for the karma engine (another world of collision physics), or karma being disabled for those static meshes.
1. StaticMesh'myLevel.Mover1.sqrmover1'
(This is the one used in the room where you can teleport to the main room.)
This one already had a collision model and thus a karma primitive, but the static mesh mover added in the map had Collision.bBlockKarma = False, so it would be ignored in that case. I set it to True.
2. StaticMesh'myLevel.Movers.squaremvr3'
(This is the one near the area with the moving door.)
No collision model existed so I made a collision model (unrealed makes a karma collision from it) for it then set Collision.bBlockKarma = True for that static mesh mover used in the map.
3. StaticMesh'myLevel.Movers.squaremvr4'
(This is the largest one, used in the main area.)
The same as squaremvr3.
If it doesn't work now, then you've probably tried it all already.

Since I've already done the changes that are required (and rebuilt the map), I'll just put it here until you get it.
Results: If my player would die/suicide on these movers before these changes, the player character would fall through it, because the karma engine had no collision model to use.