As others have said...if the pelt is prime, then skin, flesh, stretch, and tan it.  Otherwise, they lay where they land.  As far as trapping goes, I used to trap the heck out of them in the early '80's, and for some reason, there were a lot of deer around then too.  Now, with the whole "live trap" bs...well, I don't know how to convince a coyote walk into a cage.  Was hard enough just trying to make them put a front paw on the center of a pan, or run their head through a snare.  I know of nobody who puts up much coyote fur with a live trap.  Also, for some reason, the deer population isn't as high now as it once was.  I'm sure that's just a coincidence.