Once the main body of our box is all dovetailed together, I usually go head and cut out the handles next, starting with an auger, then rasping and sanding it to a hand-friendly rounded edge.
Then, we get to a new process. Since we want the bottom to be able to expand and contract with humidity changes, we need to put it in a groove. This is basically the same process required for a drawer, though in this case we have to put it in as we glue up, rather than later. I use a 1/4″ groove, set 7/32″ deep, and up 7/32″ from the bottom. I don’t usually use 32nds, but that’s what comes out of “between three-sixteenths and a quarter”. My plow plane (which we modified recently) is an old Stanley 45. As our stock is pretty thin, the fence will project under the workpiece, so we have to hang it over the edge of the workbench a little, just enough that the fence doesn’t hang up. Work straight through both the sides and the ends. If you remember, we had to leave our bottom pin on the ends 1/2″ wide. Here, you can see that’s to make sure that the groove comes out the end of the pin without interfering with the dovetails.
Grooving is pretty straightforward. Each side should take less than a minute. Just be sure to keep your plane straight. If you have to groove against the grain (it happens sometimes), then there’s a couple of ways to accomplish that. First, take a very light cut. Second, score your groove first with a mortice gauge. If you can get the walls scored in, then the light cuts with the plow should work. It might leave a rougher surface behind, but it should work. If it still doesn’t work, you can always use a router plane with a 1/4″ blade and/or a chisel. There are many roads to the same objective.