If you did a the same gearing on top so the horse would go in a slightly elliptical pattern reminiscent of galloping. Otherwise you’d need a pin on the end of your “horse lifter” shank thingy, to connect to the pole. Or the horse will rock instead of just rise.
Course we might just be over-engineering a thing by either point. Cause then you’ve got to scale everything up to eyespec, figure out how to make the axels turn when the whole carousel turns. Gears on gears and material tolerances.
I was trying to keep it simple by just throwing some magnets in the base with repelling magnets on the bottom of the poles. So when the poles went over the magnets it would force the poles up, and let them drop back down in spots there weren’t any. But as stated this has been tried, and failed to amusing results according to @Jules.
As I have noted a bit of bamboo stuck through the side of a gear will rise and fall in a sinusoidal wave if all you are looking at is the vertical motion, but as you point out there is also the same wave pattern 90 degrees off going side to side, and if the gear is travelling along a rack there is that motion as well. you could get only the up and down motions by letting the pin slide on any side to side motions, but then you get drag.
With the strips of Soda bottle plastic from this gadget you get a very low drag surface that is both strong and springy and can be used to only apply the vertical vector. There are several possible ways to do this. I would love to find that plastic in thin sheets that rings could be cut and even as the gears went round would provide only the vertical component of the movement