The normal cut/copy/paste/duplicate commands work on nested operators, too. If you want to duplicate a part of it, simply highlight the layers you want and duplicate. If you want to move something out of a nested layer, cut the item, then click on the destination composite and paste.
You can always create additional layers within any composite by using the New Layer from Operator command. This has a benefit over copy/paste of creating an instance of the operator you choose (if it's already used in a composite).