I do not profess to be an expert on this topic. That being said here is some useful information.
https://www.fws.gov/birds/policies-and-regulations/regulations/migratory-bird-hunting-regulations.php taken directly from the federal regulations
§ 20.21 What hunting methods are illegal? Migratory birds on which open seasons are prescribed in this part may be taken by any method except those prohibited in this section. No persons shall take migratory game birds: (a) With a trap, snare, net, rifle, pistol, swivel gun, shotgun larger than 10 gauge, punt gun, battery gun, machinegun, fish hook, poison, drug, explosive, or stupefying substance; (b) With a shotgun of any description capable of holding more than three shells, unless it is plugged with a one-piece filler, incapable of removal without disassembling the gun, so its total capacity does not exceed three shells;
You would want to check the specific regulations in each state. Arizona for example lists the only legal means of harvesting migratory birds as follows:
To take migratory game birds, except Eurasian Collared-doves: a. Bow and arrow; b. Crossbow; c. Falconry; d. Dogs; e. Shotguns shooting shot
So it is illegal to harvest migratory birds with an air rifle in Arizona even though the Federal regulations do not specifically prohibit the use of an air rife to hunt the same birds.
All upland game birds and Eurasian collard doves the legal means of hunting are:
To take all upland game birds and Eurasian Collared doves: a. Bow and arrow; b. Falconry; c. Pneumatic weapons; d. Shotguns shooting shot, only; e. Handguns shooting shot, only; f. Crossbow; g. Slingshot; h. Hand-held projectiles; and i. Dogs
In Arizona to the best of my knowledge there are 3 different types of doves / pigeons which can be hunted that are considered migratory game birds: Mourning doves, white wing doves and Band-tailed pigeons. There are several other species of pigeons and doves in Arizona, some can be hunted others not at all. Rock doves / pigeons are not protected by any regulations and are considered an invasive species along with house sparrows, European starlings and Collard doves (of which there are actually 2 different species Eurasian and African) there are not limits to the means of hunting these birds nor limits on numbers taken. There are also Inca doves and common ground doves here that are protected species and they cannot be legally hunted by any means. In short be sure of what you are shooting at and if you are unsure about any of these things in your state a little time spent on-line with Google or Bing as your partner you should be able to find the information you need. Hopefully this information is of some use.
Also as always a hunting license is required to do any kind of hunting. Yes even if it is "pesting" because the rules specifically list these species and how they may be hunted / harvested a liscence is required. You will be "hunting" after all. Similar to going fishing and only doing catch and release or for fishing for species that are invasive. You still need a fishing liscence. You are out shooting animals / birds, you are hunting. A liscence is required.