In the United States, it's frowned upon to kill the fox. It's just an excuse to drink and dress up and then get on a horse and keep drinking while dangerously running through wooded and hilly areas. We realize that foxes have a natural order and take care of many vermin species for us, and so the goal is to chase the fox without letting the hounds get within killing distance. It's even more exciting when the hounds get on a coyote trail because those they do kill and everyone gets to have a fun story at the post hunt drinking party.
Basically we're the more educated and empathetic hunters unlike those snaggletoothed barbarians across the sea.