@farmerman,
Nearly all my aunts and uncles (mother had had "dozens" of such cousins) did so, too.
n the German language the following 'Kniereitvers' (Hoppe, hoppe Reiter) is most popular (and such is still done today):
Hop hop rider, when he falls then he will cry,
If he falls into the ditch, the crows will eat him.
If he falls into the swamp the rider will go plop!
There are verses that have one or more lines added, for example:
Hop hop rider, when he falls then he will cry.
If he falls into the ditch, the crows will eat him.
If he falls into the hedge, the snails will eat him.
If he falls into the green grass, he will get wet.
If he falls into the water, he will get even wetter.
If he falls onto the stones, his legs will hurt.
If he falls into the swam
While playing and reciting the verses, the child sits on a parent's/uncle's/aunt's ...knees or lap facing the parent. The person moves the knees up and down, imitating riding a horse. The parent holds on to the child's hands imitating holding reins. When it comes to the part where the parent says "the rider will go plop" the parent opens the legs or knees so that the child slips down towards the floor while holding hands the entire time. The parent will then pull the child back onto the lap.
This little acting game is a lot of fun for the child, anticipating the 'fall' from the 'horse' and being saved by the parent, staying safe and sound.