Whew, you really scared me for a minute!! I would never want to recommend something like that if it wasn't safe, so I went and looked it up. (I had received that recommendation from a veterinarian, and because of that hadn't verified it before I passed it on.)
I found multiple references for using both lidocaine (a short acting local anesthetic, commonly used as an injection nerve block or as a topical gel, the brand name is Zylocaine) and bupivacaine (a long acting local anesthetic, used as an injection nerve block, and sometimes more creatively) and benzocaine (a medium-length acting local anesthesia, used as a topical gel, the active ingredient in Orogel) in chickens, ducks, raptors, and "most species" of birds. These products have been used for traditional nerve blocks, injected into joints for arthritis, injected around wounds to provide a "ring block," applied as a gel to small wounds before suturing, and I even found a reference to mixing bupivacaine 50:50 with DMSO and applying it topically to chicks after debeaking (must have been a research project, because I don't see the factory farm industry doing that). There were no references specific for turkeys, but there almost never are, for any drugs. I have been told personally by Dr. Bruce Singbeil that any drugs that are established safe in chickens can be used at the same mg/kg dosage in turkeys safely (as long as the low end of the dose range is used because turkeys are larger than chickens, which is a basic principle of any dosing decision). Dr. Singbeil is someone I trust and is the only veterinarian in the country that is board certified as a specialist in Poultry Medicine (as opposed to a specialist in Avian Medicine, or a regular veterinarian who is licensed to practice on all species, but isn't a board certified specialist in anything) that actually has a hospital and treats individual pets, as opposed to all other poultry specialists who work for industry, universities, or public health.
So bottom line, it is safe to use "cain" medications in poultry, as long as it is done carefully. You're not going to get into trouble with topical medications on a prolapse unless you squeeze the whole tube into the oviduct after you've replaced it back inside, then hold the vent closed long enough for too much of the drug to absorb into the bloodstream (it would probably take a very long time). Just applying a smear to what's hanging out, waiting 5 minutes, then working the oviduct back in will give very minimal absorption into the bloodstream immediately, as most will absorb into the tissues. It will take hours for the drug to slowly work its way out of the tissues, giving the body plenty of time to metabolize it. If you were to use injectable local anesthetics, then poultry are somewhat sensitive to them by comparison to other species, on a mg/kg basis, so total injected dose has to be monitored carefully to avoid complications. But that's true for all small animals and humans. The only time it's not an issue is with cattle and horses, which are so large that it's almost impossible to overdose with those drugs. So I can see how many people would believe that "cain" products are dangerous in poultry. Most vets that come to farms are large animal vets. If a farmer were to ask a large animal vet to treat a 5-10 lb chicken, when he's used to treating a 1,000 lb animal without concern for total injected dosage, I could easily see an overdose occurring. That happens a few times and all of a sudden the medication is labelled as dangerous.