horseback riding and babywearing...

It would partly depend on the carrier you're using. I would be concerned that the repetitive motion would eventually loosen up the straps of whatever carrier. For instance, the baby/toddler might be very secure initially, but eventually after the motion of the horse over and over and over, things might loosen up.

It's up to you and your confidence in the carrier you're using and your ability to keep it secure.

My knee jerk reaction is that NO, that isn't safe. Especially, especially for someone who doesn't ride. Maybe you could go with friends and alternate caring for the child.

I know they wont let you but I would definitly say no even if they would. You just never know when a horse might act up or get spooked and it is bad enough to get bucked off by yourself much less with a baby attached.

I would also think the repetetive bouncing could potentially injure a small baby. The bouncing is enough to make you sore afterward, even when going slowly.

That's why you ask, right?