I have flown all over the US with 2 shotguns in a pelican 1750 case, and later moved to using a hardsided golf club case to mask the contents. The one thing I can tell you is that EVERY SINGLE AIRPORT handles the situation different ... EVERY ONE OF THEM ....no matter what airline you are flying.
In ALL instances the only constant is the need to open the case, sign a declaration the gun is unloaded, slip it in the case and shut it. The person working the desk generally has no idea what is going on and in most cases it is a scared to death women who wants the ordeal over with as fast as possible....from there it is a crap shoot.
I've been in occurrences where the case goes right down the conveyour to where bags go and that is it. I've also been in situations where that happens and then 3 minutes later 2 police officers come treat you and the weapon like your an ISIS cell member (seriously...Providence, RI). In that case the case was inspected, blah blah and I got to "watch" from behind a sheet of glass.
Other times I've arrived at my destination to find the gun case has been inspected somewhere along the way, TSA puts a card in the bag / case / etc if they inspect it, and WITHOUT TSA locks on there they would have either cut off the lock or worse detained my case since they couldn't properly inspect it.
I've also arrived before, when flying with 2 other buddies who had inferior cases, to find 2 of 3 cases destroyed beyond recognition....clearly done on purpose. They did manage to break one of the handles off my pelican.
Which brings up another point....EVERY SINGLE AIRPORT IS DIFFERENT in how you retrieve your gun once at your final destination. Some places it comes out on the same belt as regular luggage others it is locked up and you have to show your baggage claim ticket.
My advice is to
A. buy a hard side golf club case and use it instead of a dedicated gun case...you can also fit a lot more stuff in there with the guns which is nice.
B. know that you CAN fly with ammo and weapon in the same case
C. Use the TSA locks...2 of them on the case and take 2 spares with you in your carry on / normal bag.
D. Expect shit to be different and a mess everywhere.