All of this is very seat of the pants and trying to learn a little each time. This is just my first attempt. Mistakes were made. Successes were had.
My main concern was whether the dies would stand up to the pressure of the plunger. I used Valentine Clays B17C-grogged stoneware because it's what I happened to be using for a different project at the time. I rolled out slabs and marked circles with the end of the clay gun barrel. I cut slightly larger than the mark to account for the shrinkage of the clay.
The inner shape is set back a bit, not flush with the outer ring of the die. The clay has to go around the supports inside and re-join on the other side, so this, I hope, will give it a better chance of doing that.
There is one more die I made, but I'm going to try it another day. Possibly with nicer clay.