Anyway, here's the initial construction. This is three layers of foam core with a layer of chip board at the bottom. Circles are cut in the foam core; that's where the figures will be placed. Each layer of foam core is thick enough that the bases of the figures should sit in there snugly and be somewhat protected from being accidentally knocked about.
For the next steps, I'm going to wrap the back and thickest part of the sides in paper to hide the rough edges of the foam core and chip board. I'm also going to trim the front edges of the foam core at an angle, so that it's less of a hard edge and suggests a gradual incline.