I guess that one of the tins will contain the music-box proper, and the others will just need to revolve. Or might the tins be empty with the tune device below the turntable? That would make sense from an engineering point of view, as the weight of the clockwork will otherwise complicate things.
One fairly basic way to 'drive' the tins would be to mount each one on a vertical pointed rod resting in a dimple inside the tin lid. You would need a big hole in the base of each tin. The top would be cut into louvres (like a fan shape) and a single small candle burning in each one would create a draft, turning the tin. In this case, you would not need gears, or a handle, etc, but you would need skill in cutting the thin steel of the tins. And there would be a distinct fire risk if the device was moved or jolted while in use …
If you can make the base thick enough to hold the music box, it could also hold a small electric motor, run off a small model battery. Like the ones used in model drones but less powerful. This could drive a series of small, vertical spindles, rotating fairly quickly, with the tins each driven by a rubber band round the tin and the spindle. The difficulty would be in mounting the tins so that they could rotate freely but not be pulled to one side (etc) by the drive.
Perhaps you could advise us – how much skill do you have in such things? Or are you asking if one of us might do the work for you?
Regards, Tim