Whether they should do it or not according to company policy, it's a YMMV situation as it can and has been done. I've done it a lot.

I've had AA reps, DL reps and UA reps "short-check" me so I'm checked in only for the domestic segments on ticketed itineraries for international travel so that I can pick up the passports/passports with visas, or other international travel-enabling docs at the transit cities that are my gateways out of the country. Other times, they don't even short-check me, and just check me in all the way but without the Docs-OK stuff. Either way, there's often that "travel docs" check beep that airlines get when scanning the boarding pass for the domestic flights feeding the international flight on the PNR.

The airlines should be doing the travel doc check at the gateway for the international flight out of the US since people really do lose/misplace passports after checking-in for and/or boarding the domestic flight en route to the same-day international flight on a ticketed PNR. The passport checks for the domestic flights feeding an onward international flight are a courtesy for the good of most travelers, but I can watch out for myself without needing the airline doing it for me.

Because they don't want passengers who may have forgotten/lose/misplace their passports before getting to the gateway airport for their international flight out of the US to end up in limbo at at an airport being transited.