Ah thanks! At first I was trying to copy the objects from the project sidebar, but now I see I actually want to copy the contents of a layout or the contents of the event sheet itself.
That seems to work pretty well. One problem I ran into is that if the event sheet relies on any of the system objects (eg: mouse or audio) then I need to manually add each of those.
But this copy and paste is at least better than nothing.
Thanks for your help!