My dogs do not kill things -- I do.
There is not much skill in killing, but it seems to be the thing that fascinates non-hunters.
I terminate quarry with a blow to the head -- the same as is done with slaughter house beef. It's fast, it's assured, and there's no chance of shooting a dog or anyone else. Only after a groundhog is good and dead do I let the dogs rag the carcass, as can be seen here. In this case, it makes for a dramatic picture, but in fact this groundhog is as dead as a donut. Dead is dead. The reason that this groundhog has blood in its mouth is that its skull is in pieces insider its own skin.
There is no coming back for you after I sort things out.
Let me stress that dispatch is an option that should not always be used.
If the quarry is a fox or raccoon, serious thought should be given to letting the animal go so that it can be hunted again another day. A farm can easily be hunted out, especially if you are hunting more than a few times a year. Having said that, an animal that is wounded should never be released. It's better to terminate a wounded animal than have it suffer infection or starvation in the wild.
An alternative to dispatch is relocation. Animals that are tailed out alive, snared, or netted can be placed in a fiberglass bag or large Havart cage-trap, or small dog box kennel, and moved to distant farms or wildlife management areas. This is often illegal in the U.S. because of the notion that rabies might be spread. Good point, but total nonsense in this area, where rabies is endemic.
If you decide to dispatch, or the farmer requires it, there are three options. The best, in my opinion, is a hard rap to the skull with the back (blunt side) of a machete blade or some other reasonable weight. A strong well-placed blow with a dull instrument will create instant death and no pain, and very little blood will be evident. It is also very safe and has no legal complications. This method should only be used if there are no loose dogs, however. A blow to the head can easily be delivered to a terrier if it is still jumping about and leaping in trying to rag an animal that is pinned under a boot or held in a snare or net.
A common dispatch method is a .22 round to the brain-pan. Without a doubt this is a very quick and humane way to dispatch an animal, and a very safe one too if the shot is done in the hole after the dog is pulled. Some farmers are a bit dodgy about guns on their properties, however, and anyone with a .22 revolver should be sane, sober, and not very excitable. If you are not familiar with guns, there is no reason to buy one now to do terrier work -- a strong rap to the head will do the trick.
A third method of dispatching quarry is with a large knife to the heart, or to the neck vertebrate, with another cut to the carotid artery. This method works well if you have loose dogs about, as a knife can be methodically placed through the heart, or driven into the base of the skull severing the neck bone.
A final tip is that a heavy chisel-tip digging bar aimed straight at the head can take out a groundhog that is poking out of a deep pipe. Aim well and hit hard, and it's instant death.
Never let your dog rag life quarry to death -- it is a sickening thing and debases not only the person that allows it, but all working terrier enthusiasts.
After you have dug a sette in a hedgerow or forest, please take the time to repair the den by jamming sticks cross-wise into the hole so that as much of the den pipe as possible is preserved after you backfill it with spoil. A reconstructed den will, in time, be recolonized by raccoon, possum, fox or groundhogs. Reconstructed den pipes help guarantee productive hunting grounds for seasons to come.