If mixing calibers, the easiest and most relevant factor is FPS. At the same fpe, the .177 will be more apt to penetrate than .22. They are more similar in penetration at the same fps.
Head type is also important. Flat nose (wadcutters are less likely to punch through than domes.
Lead less so than alloy.
Fortunately you don’t need much to take down birds cleanly. Inside of 20-30 yards, a wadcutter at 500-600fps is plenty. The trick isn’t finding an accurate wadcutter at those distances...inside a dime, for example. In my experience, the RWS Meisterkugelns and often times even the Basics fit that description.