When the overlay is first connected I can move it fine. When I later come back to the overlay I am not able to move it anymore; the drag arrow is gone. I have to delete and reconnect just so I can move it again.
I can't reproduce your glitch, I am able to move overlat when I am switching between Design and Prototype modes.
There can be situation that do not allow to mcve overlay. That happens when your overlay artboard have the same size as target artboard. To solve this issue please change your overlay artboard size. If it not possible you can relocate objects in overlay artboards for moving them on target artboard.