Just for clarity (and a bit of geekery) the amount of kinetic energy necessary to kill a rabbit with a headshot is around 3 or 4 foot pounds.
In England and Wales the maximum muzzle energy for an air rifle available for purchase to a person over the age of 18 without a firearm certificate is 12 ft/lb. Ballistics dictate that a pellet fired from a air rifle loses energy over distance, and as stated above, the trajectory curves and drops as distance increases. This trajectory curve becomes more pronounced in heavier calibres, but even at distances of fifty to sixty yards, the pellet from a modern rifle will still carry sufficient energy to kill cleanly.
Many of today's air rifles are supremely accurate, especially precharged pneumatic rifles, which tend to be free of recoil or mechanical lash. They also tend to be equipped with high magnification scopes and in the hands of a reasonably skilled shooter pellet placement is consistent and accurate even over distance.