on thing for your guys to remember is that shot gun patterns are taking on the appearance of group shooting.
Shooting at running coyotes at 25-35 yards and having a shotgun pattern the size of a dinner plate or smaller will leade to a lot of misses. Few remember to leade a coyote when he is running or trotting, they appear so large.
Also, when shooting at a pattern board, many will forget that often you will knock them down and a follow up shot is needed...keep that in mind.
So, keep in mind the terrain you hunt in and choke accordingly.
Most of my hunting with shotguns is in dessert terrain, rarely did I ever have to shoot a coyote beyond 45 yards. ON the other hand, I did have to shoot a lot with them hauling azz in at warp speed with shots being between 5-30 yards. I recently sold a Rem 1100 that had killed over 500 coyotes with a fixed full choke, back bored, forcing cone lengthened. Guys that hunted with me loved that choke because it patterned "just right". VERY few coyotes got away, with 3" lead BB's, Rem, fed, or Win, and factory #4 Buckshot.
Later on, I got screw in chokes for my Rem 11/87 shot a Carlson's turkey choke. It takes quite an experienced shot gun shot to kill coyotes with this gun as the pattern is 15"-20" at 40 yards. I quit letting novice shot guns shooters use the 11/87 with the turkey choke.
3-4 pellets of #4 buck and the coyote is on the ground, keep shooting till there is no movement!
Over the years, I took a lot of different guys calling. I tried all kinds of shotguns, from the very light Ithaca 37 to the heavy weight Ithaca 10ga semi auto with it's 32" fixed full choke, lots of 870's with different barrels, BPS, Super black eagle, Mossburg 500's, 1100's with barrels from 32"-18", beretta 390/391's. Guys that do not shoot shotguns a lot or little at all will do better with a softer kicking shotgun with a std full choke with lead shot that is not a chunk of lead to handle. It has never ceased to amaze me how many people can not hit a running, trotting, walking coyote with either a rifle or shotgun. On a trip to Mexico in the 80's, one guy missed 12 coyotes with my shotgun. Later on, we spend quite a bit of time on the skeet range, he did not miss any after that!
So, the word of caution, balance your shotgun choke with your terrain.