My dog is like that, when we go out in the pasture he follows us sort of, sniffing around the brush and brambles. All of a sudden you'll hear a yelping, baying bark, and he's off on the chase. Never seen him get a rabbit that way, and squirrels just antagonize him, too. But it's good exercise for him, I guess.
But, sometimes, he'll find a baby bunny somewhere, and bring the carcass and plop it on our deck. Nasty! I have to flick those off onto the grass. I've seen him take one of those carcasses and play with it, like grab it and throw it up in the air enough times until the head flies off! Gross! What a great hunter. Not.