I'm new here, but I don't think I could excise reference to the V-22, nor would I want to. I just think we're going down a rabbit-hole that mixes two separate topics.
Marine V-22s don't train for SAR, they train for TRAP. TRAP is going in force, with escorts and a unit of grunts to get a survivor and/or aircraft down in a known location. We don't practice that search pattern expanding square, creeping line, and the rest of that crap. Other than the station assets in Yuma and Cherry Point, no substantial time is spent training or doing SAR work.
As far as the USAF, they have dedicated platforms that perform CSAR in theater and can perform stateside SAR for the military as well as aid civilian authorities periodically. I think some of the ANG squadrons, especially, spend a good deal of time aiding the civilian side of the house with that. Others can give more fidelity on that.
Regular military assets do get tagged periodically to do SAR. Sometimes that is as simple as ATC saying,"I lost radar contact with an aircraft at XXX, can you fly over and check it out?" If it's something more substantial than that, it will involve deliberate planning, and the authorities will go first to the units best prepared for it, e.g. USCG (obviously), USAF SAR, ANG, Navy HSC squadrons, etc, etc.
True SAR involves specialized training and often specialized equipment. Not every jerk with a helo and a hoist can do it. I'd be happy to take on the duty, just give me 50% more pilots so I can cover the graveyard shift every night and 50% more flight hours so I can train for it and know what I'm doing.
I don't want to be over the water at night, paging furiously through the SAR TACAID checklists, going,"S**t! How do you figure out the search area again? He punched out there, the windspeed was 180/5, the ocean current is SW at 3....carry the one...multiply by 1.5...take the square root of my birthday..."