A strange thing happened a few years ago when Christine Longnecker, who teaches horseback riding in and around Erie County, Pa., brought her new rescue dog, Miles, to a class. Instead of waiting quietly with the other non-horses in the barn, Miles suddenly sprinted into the ring and bounded over the fences himself.

“He looked so excited,” Ms. Longnecker said. “And then he turned and barked as if to say, ‘This is how you do it.’”

That was the beginning of Miles’s career as an agility dog — the sort of dog you might see sprinting over and through obstacles while its owner frantically rushes around, yelling commands. On Saturday, he is scheduled to compete for the second time in the agility competition at the Westminster…