For now, I'll use whatever I'll find in my garage or basement. You mentioned this previously:
Am I right that I can use the gear motor from this rechargeable battery powered screwdriver? It says Type 1 2.4V and 150/min RPM. On the other side, it says NOM 2,4v.
150 rpm is still going to be too fast for you I am pretty sure. If you used threaded rod the speed would probably end being a foot a minute or somewhere in that general region. I suppose you could turn it on/off/on/off etc if the spacing between your photos doesn't need to be super-precise.
But it would be easy to use that in a direct connection to the threaded rod. Just take two nuts and turn them against each other at the end of the rod and put a socket adapter on the screwdriver. At the other end of the rod you will need a bearing. If you had a small wheel sitting around you could use that. If not, Harbor Freight sells those types of thing for dirt cheap.
Lastly, you need the carriage that attaches to the sled. Here it would be nice to have a welder. It would be very easy to weld a nut to the end of metal bracket that attaches to the sled pulling it along. With no welder, you might take a piece of wood, drill a hole that is just a touch smaller than the outside dimension of the nut, and pressure fitting it (ie hammer it in) with a bunch of epoxy to help hold it. Then the wood bracket can attach to the sled. For a more stable carriage, you could use two nuts spaced an inch or two apart that attach to the same carriage.