The problem is the object moves fine but it's collision data stays put, along with the player.
I've tried messing with collision import options on the parent object but those don't appear to do anything, at least as far as the collisions staying put. I also tried changing the BSX flag on the .NIF itself to 3, and several other settings, which just appear to make it either not have collisions at all or the game to crash.
I'm sure it is some simple super-secret flag hidden somewhere obvious, but looking at other moving collisions in the game I am at a loss to see where to enable this.
Perhaps it is more difficult than I imagine... Somewhere I read something about creating an invisible collision box around the player and moving that, but would you run into the same problem?