Good post, Anne.
In answer to the original quesion “Do the hounds kill the fox/coyote” the answer is yes, if the pack catches it, they will kill it. But as in so many things, the little word “if” covers a whole lot of territory.
In the U.S., if the quarry goes to ground, it is not dug out, terriers are not sent after it, it is given “best” and hunters hope to encounter it another day. In the U.K. (I have not hunted there, so this is second hand), the landowner expects the hunt to perform pest control service and kill the fox. So earths are stopped (to keep the fox from going to ground) and terriermen are used to dig out the quarry and dispatch it if it does go to ground somewhere.
In the U.S., the hunt usually ends when the hounds lose the scent, or the quarry goes to ground, or runs out of country. When the quarry runs past a fence line or across the road where we no longer have permission to hunt, hounds are stopped and the huntsman takes them away to look for another line somewhere.
I have twice in the more than fifteen years I’ve been hunting seen an individual hound catch up with the quarry. Once a red fox, once a gray. On the first occasion, it was a young hound and a young fox, and they literally played together like two pups meeting in the park. They frisked and rolled and tail-wagged and bowed and tumbled together until some older hounds headed over towards them and the fox exited stage left and went quickly to ground, completely unharmed. The second time, it was a gray fox and a seasoned hound who had stayed the line when the rest of the pack missed a turn. So Famous was out in front by a good lead, and the gray fox had stopped on the trail to watch the festivities. Famous caught up to the gray, the stood there and looked at each other, Famous took a step forward, they sniffed noses, and they just looked at each other until the pack found the line and caught up…about 90 seconds later. And then the fox thought maybe this was a fine time to boogie and off he ran, and Famous and his packmates chased him and the fox darted into a thick cover, hounds lost, and the fieldmaster saw a gray up in a nearby tree. So IME the quarry needs to be caught by the pack to be killed, and, frankly, that is quite rare. I personally have only seen hounds catch up to quarry two other times in all the years I’ve hunted with various packs: both times the pack killed the coyote, one was very mangy and sick, the other was healthy and either very stupid or very unlucky.
Here is a chart on animal speed. http://www.factmonster.com/ipka/A0004737.html Note that the coyote is faster even then the grayhound and whippet and certainly faster than foxhounds. Plus, the coyote has tremendous stamina and can maintain speed for a long time, as opposed to a sprinting animal like a quarter horse who has a quick afterburner but not a long-distance burn.
Interestingly, google just popped up this webpage for me http://www.imbmonsterbucks.com/predator.htm, showing what’s done by people who buy furbearers licenses and go out yearround to kill coyotes with rifles. They use prey calls to coax the coyotes into range and shoot them. They quote stats that the calls bring in a coyote 1/10th or 1/3rd of the time, depending on where they are hunting. And then the shooter fires. This is a very different kind of hunting than mounted foxhunting, to say the least. We don’t lure them in, we go out into their environment and using only the noses of hounds we try to find where they have been and hope to catch up and get a view. The coyote are out there in number and being hunted year round by rifle hunters and also being trapped—neither method insures the all-or-nothing outcome that a pack does. The shooter can wound and leave the animal to slow death. The trapper can maim and leads to a very slow death. The hounds either kill very quickly or (much, much, much more commonly) the quarry gets away completely untouched.