Of course, copy and paste is supported as usual. Shapes can be copied between layouts: by opening two layouts, shapes can be moved from one layout to another. The shapes are mapped to the same layer than they have been on in the source layout. If a layer does not exist yet in the target layout, it is created.
Shapes in the selection are simply copied to the clipboard in the way they appear in the current cell. This means, if the shapes are pasted into a different layout they are put on the same position, but flat into the current cell. This provides a way to flatten a hierarchy: choose "hierarchical selection mode" in the editor options dialog (deselect "top level only"), select the shapes to flatten and copy everything to a different cell.
In non-hierarchical selection mode ("top level only" selection mode) or by clicking on a cell frame when the hierarchy levels are limited, instances can be selected as well. When copying instances to the clipboard, two possible methods exist: