I grew up hunting coyotes with my Grandpa and his Greyhounds. As mentioned you now need special permits in Alberta. I actually saw a hunt occur in Sask this winter as I was driving by.
Grandpa had a dog box rigged up with a string to open the doors. I seen him run up to 4 hounds, but generally if we had 4 we ran two pairs.
We drive around till we spot a coyote where we had permission, then turned them loose. We didn't have any special kill dogs, all the grey hounds we had were quite capable of killing coyotes, occasionally even 1 on 1. I remember once releasing the three hounds we had, and getting 2 coyotes, two dog's took one, and the other dog got the second one.
With greyhounds it was generally a very short chase, well under a mile, usually under a half mile, if the coyote had enough head start to get farther then that Grandpa wouldn't let the hounds go. There was the occasional exception when a coyote led them through the bush a ways, or was exceptionally fast.
The best watching I ever saw was when we turned em loose on a fox on a warm day in early March, about half the snow was gone on a summer fallow field so there was about an inch or so of mud. Every time the hounds got close, their would be a ball of mud and snow, and the fox would be 20 yards to the side running hard. Seems a fox can corner much better then a grey hound. The fox ended up getting away unscathed, but it was still well worth the price of admission.
Oh, and to those who were talking about mean dogs, I don't think that would have been greyhounds, all the greyhounds we had were very friendly, and not just to us, they were generally good natured dogs. In all the years Grandpa had dogs, I only ever heard of once he had one that bit. It bit my dad and Grandpa didn't believe it, till it bit him, and that was the end of the dog. However that was once in probably 40 years of running dogs, I never did see a mean one in my 15 of being with him.