Hi CBWW
Looks like your on the right track. The few I have worked on had slide/ runners attached to the inside of the opening. Looks like the one in the photo has the dado [ slide ] cut into the compartment sides. The last one I had, had 1 1/4" wide x 1" thick board running from front to back on each side with a 3/8" dado cut just below center running the length of the slide. The mirror frame was attached to a 1" thick x 2 1/2" board with tenons cut into the ends to slide in the runners. The slide and runners are set even with the case . Now for the tricky part, The board the mirror hinges attach to is cut to slide over the back of the case. A notch is cut into the back of the case 1" deep and just up to the slides, preventing the mirror hinge board tenons from sliding out the back. The mirror hinge board is notched so when it is slid to the back side it extends over the back boards of the case. The back boards prevent the mirror hinge board from sliding out the back, and the mirror hinge board is notched so it slides even with the back of the case. The hinges are set on the rear edge of the board and a bevel is cut on the mirror frame to allow it to set at a slight angle when open. The mirror is able to slide from front to back of the case and will lay flush with the top when laid down. Sorry , no photo's , I thought it was so simple so I didn't take pictures. Hope this makes sense. The piece I am describing was French about 1800 - 1820 and gesso with painted women in gowns of the period on the lids of the compartments.
Randy