coyote = .223/257
deer = .243/30-06
elk = .270/.338
griz = .338/.416
I consider lesser than stated mins would do the job, just not as reliably. Higher ratio of wounds, lower ratio of kills.
Greater than stated max's would probably result in more misses, due to greater recoil/muzzle blast. If you don't hit the kill zone, it doesn't matter what diameter, velocity and/or bullet weight you are using.