Remember that the biggest air rifles have about the same ft-lbs as a .22 Magnum - though better long-range ballistics due to the larger mass involved.
Let's say something special like a Quackenbush or Barnes or other high-power 50 cal or larger model might be hot-rodded to about as effective at longer ranges as a .22 Hornet.~650-700 ft-lbs) That's still really pushing it and most serious hunters still wouldn't recommend such a small round.
This last season,With one .495 roundball I took my deer (~200lbs) at 30 yards and he ran about 50 yards afterword with a stock Dragon Slayer. Look up my post in the hunting gate for more specifics.
Two people missed w/3 shots each using 30/30s.
One person fired 5 times with a .270 and hit the 120lbs Doe 3 times before it went down. The only meat we got off that deer was off the rump.
The choice of bullets goes 2 ways. Flat nose like Butcher likes and hollow points. Hollow points don't work without speed and a lot of big bores don't have enough speed to make them open properly. If it won't shoot over 700 fps, use the flat nose. Velocity translates directly to distance, so a rifle with higher velocity will have a longer effective kill range, too. I know guys that have used a .308 air rifle to kill deer. They shoot head shots with 125-150 grains and are inside 50 yards.