Don’t let your dog run over to me.
Don’t let your dog run to me.
Don’t let your dog run towards me.
What are the differences?
“Run to me” and “run over to me” mean approximately the same thing. “Over” is redundant but emphasizes the direction of motion.
“Towards me” means “in my direction” but does not explicitly mean that the dog will reach the person. However, in this context the person is most likely concerned about the dog reaching them, so the destination is implied.