Most of the time rabbits "circle" or end up where they were jumped at, deer tend to also but they run a much, much larger circle or route. Knighthawk could probably explain how deer run better than I, it is not legal to hunt deer with dogs in Michigan (I always thought it sounded like a heck of a good time though). Dogs can definitely get lost chasing deer, they run a bigger risk of getting hit by a car too because deer run so much farther that they cross roads way more than rabbits tend to. Unfortunately there are some misinformed people out there that might shoot your dog if they see it running deer, a lot of beagles have run deer and most can be broke from it and it is too bad some get killed before their owner gets a chance to fix the problem. You can lose dogs running snowshoe hares too, they tend to run way bigger circles than cottontails and their habitat is usually more remote. That's why I like my Garmin tracking collars so much, I have been able to tell exactly where my dogs are all the time.
I have a beagle that kept running away. I am not using him as a hunting dog, just a pet, so I invested in an underground fence. It has been working great, except the other day we lost power and I wasn't thinking and he darted out and ran off chasing a deer. Luckily, someone found him and I got him back. I was wondering is there any way to train a beagle to stay home! Thanks!
Yes, rabbits run in a circle. If you wait by where the hound picks up the sent the rabbit will eventually come right back there unless he goes to ground i.e. ground holes, or under buildings. If you are letting them run where there are deer, especially in the fall when they are in esterise (mating) they give off powerful sent and the beagle really like it and will chase. Take the advice given by other here. Get a training collar and use it only when they are chasing deer. It seems cruel, but better than them getting into worse trouble.