Maintain selection when switching layers?

[Ofnuts]

Incorrect. The selection is on the whole image, so you can select something on one layer and move to another layer to use that selection. Your problem may be that the selection markers (aka, "the marching ants") are not visible, so they disappear after you make your selection (but the selection is still there). See View > Show selection (you can also hit Ctrl-T to toggle that setting).