At various times I've shot either deer, elk, or antelope with: .22LR, .22 WMR, .223, .243, .250 savage, 6.5x55, .280 rem, .30 carbine, 30/30, .308, .357 mag, .45 Colt, .45/70, .50 and.54 muzzleloaders and a .648" ball from a 12 gauge muzzleloader. So far as I can determine any one worked about as well as any other. No animal dropped in it's tracks except for brain or spine shots. No animal traveled much over fifty yards from where hit except one antelope hit at about 400 yards with the .280 Remington and 160 grain Nosler partition, that one walked about 200 yards and lay down. That was the only animal which required a second shot.
I used to be a fan of Elmer Keith and believed "bigger is better" but my own experience now leads me to believe that if you put a bullet in the right place it doesn't make much difference as to how big or small the bullet nor how fast it travels, they'll all kill rather quickly. And if you put a bullet in the wrong place? Well just don't do that!