I have a variant on the horse one:

(bouncing child gently on knee)
this is the way the ladies ride - trottety trottety trottety trot.

(bouncing less gently & then lifting them up in a "jump")
this is the way the gentlemen ride - gallopy gallopy gallopy and OVER the hedge.

(rocking wildly from side to side then dropping child down between the knees)
this is the way the farmer rides - cobbledy cobbledy cobbledy and DOWN in a ditch.

(Berkshire UK)