I would put them in a large rubbermaid bin (or any container big enough to hold the nest) with an escape barrier at the top. You can connect a tube leading to a new nest. They will move as it dries out, or as you pick out larger pieces that fell off.
If possible, raise the nest on some dowels and take the glass off if it is that far gone. They will swiftly move.
Thanks! That's a good idea. I hadn't thought of using dowels.