If you're in luck, then a) all four receptacles are on the same circuit. b) The first box is fed from below, and the others all connected horizontally, as your husband says.

How far apart are they? I would draw a rectangle on the wall with a line above the receptacles and another below where you want them moved. Then cut out the drywall in a neat rectangle. You will then see what is happening. You would then duplicate what exists, but just lower. If the first box is fed from above, then the simplest solution is to use it as a junction box and wire the four lower ones as above. Then put a blank plate over this junction box.

Another issue will be where the existing wires are coming from to those boxes; there can be many configurations. The wires could be coming from below, in which case you may have the necessary slack to have the wires reach the new location, or they may be coming from the top and won't reach to the new locations (and it's not legal to simply splice wires in the walls). Your husband is likely correct that the wires also go laterally through the studs; and they may be stapled at various points within the walls. You also don't mention whether the outlets are all on the same wall. If not, they could be coming around corners, going over doorways, etc., all making it impossible to pull the wires to new locations. I've moved receptacles before and it typically involves at least some pulling and fishing of new wires, and a little or a lot of drywall removal. In short, this could be a routine task, or a quite complicated one, and you won't know until the electrician starts investigating.

Lots of variables here. If there are corners involved, it is likely they will have to cut holes to turn the corners. If they are all on the same wall they can use the drill bit with a long extender to drill the holes to run the wire, they will use the outlet box holes. If the wires come from under the floor that would be the easiest solution, just cut the new holes and install the new outlets.

Either way you will have to patch the old outlet box holes. Drywall patch, mud, sand, paint.

Powermuffin, are you proceeding with this project? You never said what kind of wall surface do you have. If it's drywall, than repairing will be easy. A little more difficult if it's an old plaster and lath wall. Probably impossible if it's wood paneling without replacing the paneling. Let us know!