Varmit, I have never heard of using bulldogs but some people use the name interchangably with pit bull. To use a pit, the coyote is called in, and I mean close. The pit is released and he engauges the coyote. If more than one coyote comes to call then you will have to shoot the extra coyotes while they are attacking your dog. A good pit can easily handle one coyote, some times two. I will tell you this, the pit that is in your back yard or your neighbors is not the same dog that is used for this purpose. A PIT BULL is not a pitbull no matter what the media says. The dogs that are used are smaller than you would think, silent and make quick kills, the coyote hardly know what hit him.
The greyhounds that are used are not the slightly built track dogs. They are usually about 28-32" at the shoulder and out weight a coyote by at least 20-30 pounds. They are a substantial dog and although their top speed only lasts for about 500 or so yards, their "cruising speed" can over take the coyote. They are generally released in twos or threes and every time the coyote changes directions he has a fresh dog on him. When he is caught he is more tired than the greyhounds. A coyote can put up a good fight but two greyhounds can kill a coyote. A good working greyhound is a sight to see. Several other breeds of sight hounds are also used, one of the more popular breeds is the american stag hound. If a small greyhound is used, it is to stop the coyote until the hounds arrive.
Varmit, the dogs used for this, be it a pitbull or a sight hound are very specialized in the purpose that they serve and are bred for that specific task. Ain't dogs interesting!