I've killed deer and hogs with the .32-20......from a single action revolver!! It's not something I would choose "deliberately" to hunt such game, but when the opportunity is there, you use what you have with you.
In my experience the .32-20 (loaded with hard-cast bullets of 115 grain bullets) is in nearly the same class as the .357 from a handgun. Should work quite weel from a rifle as long as range is limited and shots exactly placed.