I assume you mean that the receptacles are now at normal height (10-12" off the floor) and you want to use them to power new undercabinet lights and receptacles near the cabinet surface.

You will need to cut out a square opening in the back of the base cabinets so you can reach the old boxes after the cabinets are installed. You will also need to install box extenders to make the box stick through the cabinet, and then install covers (either plastic or metal) to cover the box.

You could extend the wiring through the box cover but then it has to be covered with conduit or wiremold to protect the wires. If it was my house I would try cutting into the drywall directly above the existing outlet and fishing the new power wire down to the old outlet.

For old work boxes, as long as the wire is clamped at the box in the wall and at the fixture, you don't need to staple it to the framing in the wall. Old work plastic boxes have a built in wire retainer (just a springy piece of plastic), and you can also get old work metal boxes with a screw down retainer for NM cable.

You can mix plastic boxes with metal covers, and vice versa, but if you use metal boxes you must ground them to the same common ground as your cable.

Bruce

I'm not sure about metal covers on plastic boxes, it's not good practice and may be prohibited, unless you ground the cover.