Update (still a problem): Redid it without changing the pivot point. The different body parts line up perfectly in proportion with each other, like they're supposed to, but I still can't move the body. If I delete the one-piece model, and move the new parts in instead, each body part snaps back into its old position when I click it. So if I click all the body parts once, the whole body just moves to the old location.
Note: In the Modify panel, the body parts display at a different location when activating and deactivating the skin modifier. For example if I click "Editable poly" and not "Skin", the body part will move a bit to the right. When I deselect Editable poly, it snaps back onto the body since Skin now is selected.
One option could be to move the skeleton into the new position of the body, but I have the feeling something's fundamentally not in order... any clues?