How do I, without madness, move a sprite between layers?

What are you trying to do? Because the object is global and in a fixed position it is always outside of the layout.

I want the sprite to move from layout to layout, like you'd expect in a platform game when a character moves from one screen to the next.

If the player must be global then you need to set its position to somewhere reasonable on start of layout, but don't apply the same event sheet to every layout, they will need separate event sheets or another possible option is logic in one event sheet that filters out the layout such as if layoutname=this.

You don't use multiple instances in each layout, it is global so you start with one in the first layout that's all you need. I meant set position with events, on start of layout 2 : object set position to relevant x,y.

Here is an example (made in beta r168) of a character object that will come from outside of the screen, will give the hand back to the player on collision with a specific "marker" object.

The character is global and is positioned on start of the layout to a specific location.

Two layouts, using only a single event sheet.

On start and end of the level, the character is automatically moved, ignoring user inputs.