as most have said already here....easy as pie. seriously. traveled with my firearm about 5 trips this year and not a single problem/issue.
i fly southwest out of midway....at check in...i say, "i need to check a firearm in my luggage."
i don't have it out on it's own case...i have it inside my luggage - inside a secure case of course.
a pistol carrying case on it's own (pelicans, etc..) is much more likely to be theft target.
southwest gives me sheet of paper and a tag to tape to gun case (again, inside my luggage). walk over to tsa...they (TSA) have NEVER opened the gun case. they check to see if it's locked. tsa adds some highlighted text to my luggage destination tag. and off it goes. several times this happened within sight range of CPD and no issues at all.
pick up at luggage pick up at destination city.
fyi...i only bring pistol to states that i am licensed to ccw. that's just me.
WARNING - please be VERY CAREFUL if they have a stop in NY and for some weird reason have to deplane and have to re-check luggage again. NYPD will arrest you. No ifs ands or buts. very sad to hear it...but, that's their policy.