My Dad was a "pigeon" person and would do races with other people in his circle. They would release pigeons from different locations and then clock the time before they returned to their roost. We released a pigeon 120 miles from home and the bird was sitting on top of the coop when we got home.
I'm not sure if USPS ships live animals.
BTW: I've had pigeon in Europe and Egypt and it can be one tasty bird. Especially stuffed with rice and seasoned with cardamom and cinnamon.