It can move, that's kind of the point of it...
We often don't move it a lot, but it can make sense to a)move it to where you want to jump off (1 action), bring your mouse into the tower and off to a free neighboring spot (2 actions) and then shift the wall to block someone sliding a certain piece (1 action for 4).
My question has always been whether a player can go back into the three cheese block tower, and then have it moved. You can go into it from outside the castle and then move it, but as the main game says you cannot re-enter the towers, it seems like its not legal to go from the castle back into the tower and then move it around the walls.
Is it allowed to move the tower to another wall, or should it stay where it was at the beginning of the game?
If it should stay by one wall why do they say:
"Each switching of the three-cheese-block tower counts as 1 action.
During your 4 actions per turn, you can choose the action "Moving
the tower" as often as you want."
I can't imagine more than 2 moves that make sense.