I don't have a lot of information for you, but if we jump to the mid 1800's greyhounds were brought to the states from Ireland and England for one purpose and that was to hunt and kill jackrabbits. The jacks were eating the farmers crops. the greyhounds would catch and kill what they could and I don't know abut eating the jacks, but I would think the farmers didn't care if they did.
if you go back a thousand years or more coursing was a sport even back then and it wasn't about the kill as much as it was about the chase. The hare would be given a head start and the chase was on. So catching and killing the hare wasn't a priority, but I'm sure if the dog caught the hare it was dead in no time.
Even in more modern times into the 2000's coursing was still going strong in England and the rabbit always had places to escape to so the tradition of coursing was more about the chase than the kill and skilled rabbits were bred just for this. It wasn't just catch any rabbit and throw him out there.
As you may or may not know General Custer had a lot of greyhounds and other sight hounds that traveled with him. When I say a lot I'm talking 30 or 40. I know some of the other dogs were salukis which can run forever. It might be a buffalo, antelope or probably anything else and these dogs would start to chase as Custer followed on his horse. If the chase was long the greyhounds tired and the salukis would soon be continuing the hunt from the front. A pack of anything can take just about anything down, but I'm not sure if the dogs made the kill or just tired the game out and Custer was able to shoot it.
Maybe others can add to what I posted or correct something that I remembered incorrectly.
While this doesn't answer your question there was a poster here on GT for a lot of years named Martin Roper. He is very intelligent and was a pedigree guru and greyhound historian. He did a lot of research and wrote this maybe 12 years ago.
Everything You Know Is Wrong
In the 1970s, an American comedy troupe comparable to Monty Python was the Firesign Theater. They never reached the superstar status of their English counterparts, but their albums were popular on college campuses across the country. The title of one of them, Everything You Know is Wrong, has become a personal motto of mine—it seems that almost daily new discoveries in science and exploration make us reassess what we once held to be true.
The same can be said for Greyhounds. A long-held belief, published as fact in dozens of books and repeated on hundreds of web sites, is that the breed is of Egyptian or Middle-Eastern origin. The assertion is puzzling because there are no Greyhound breeds presently residing in the Middle East. The Saluki, Sloughi, and Afghan Hound are clearly of a different type than modern Greyhounds, and it’s hard to imagine that they descend from a common ancestor. On the other hand, Greyhounds and Greyhound-type dogs are common in central Europe, Spain and the British Isles. As early as 1853, John Henry Walsh, writing under the pseudonym "Stonehenge," made a clear case for a Celtic origin for the breed in his classic The Greyhound, but not too many authors since him seem to have been able to make the same connection.
From the beginning of the first millenium B.C., the Greeks were seafarers and traders and regularly visited ports all along the southeastern Mediterranean in what is now Egypt and the Middle East. Much of what we know of that area in those times was recorded by Greek historians and there is no mention of Greyhounds. The breed was completely unknown to them prior to 200 B.C., the time of their first encounters with the Keltoi, as they called them, a tribal culture from the north. In 300 B.C., Xenophon made no mention of Greyhounds in his discussion of dog breeds in his treatise On Hunting. Two centuries later, the poet Grattius wrote of the Celts’ dogs that, "...swifter than thought or a winged bird it runs, pressing hard on beasts it has found." Arrian, another Greek, but who wrote in Latin, clearly identified the Vertragus, the predecessor of the modern Greyhound.
The Celtic culture flourished from what is now Austria, west to northern Spain, and north to the farthest reaches of the British Isles and Ireland. Everywhere they went they took their dogs with them and left offshoots of the Vertragus. In Spain it was the Galgo; in the British Isles, it was a bewildering array of sighthounds in a wide variety of sizes and coats, from giant dogs we now call Wolfhounds to "Tumblers," by contemporary accounts a Whippet-sized dog. The Celts made no distinction among their sighthound varieties. To add to the confusion, English writers up until the 16th century called all the larger Celtic dogs "Greyhounds," and the dog we call the Greyhound today, the "Coursing dog." Irish Wolfhounds in those days were prized in Europe for hunting Boar, and the demand for the largest Greyhounds "of the Irish type" was great and they fetched tremendous prices.
The present Greyhounds, the ones we love, are the result of the coursing craze after the easing of the Forest Laws in the 17th century which prevented commoners from coursing or even owning Greyhounds. The coursing rules of the day dictated a very specific range of performance and traits, and those are the ones we see in our dogs today. Almost all the other varieties of Celtic sighthounds disappeared. Even the Irish Wolfhound is a re-creation of a breed that had all but gone extinct.
Two recent landmark genetic studies have confirmed Walsh to be correct. The first, "Multiple and Ancient Origins of the Domestic Dog" (Science 1997), traced the mitochondrial DNA from ancient times to the present day Greyhound. Interestingly, three other breeds derive from the same mitochondrial strain, the St. Bernard, Miniature Schnauzer, and the Irish Setter, which suggests male-line introductions of other breeds to Greyhound-line females who were the foundations of those breeds. All three originate in areas where Celtic culture flourished.
The second, and more definitive study, "Genetic Structure of the Purebred Domestic Dog" (Science 2004), used Single Nucleotide Polymorphisms (SNPs), clumps of identical DNA strings that appear in groups of breeds, but often not in others. The study clearly showed that Salukis and Afghan Hounds were part of an "Asian" group along with the Chow, Akita, and Shar-pei. Predictably, the Greyhound appeared in what I’ll call the "Celtic" group along with the Irish Wolfhound, but also as a progenitor of more recent breeds including the Whippet, Borzoi, Belgian Sheepdog, Belgian Tervuren, Collie, Shetland Sheepdog, and the St. Bernard. A glance at a map clearly shows that all those breeds originate within the influence of Celtic culture in Europe.
It’s time the Celts got their due as the caretakers of the breed, not Egyptian pharoahs or Mesopotamian kings who never saw a Greyhound.