You raise a good point, big hub airports tend to be in cities with passport offices and/or many consulates, so there could be some passengers who are making last minute travel, who might require picking up their documents shortly before their international flight.

I guess the most reasonable thing woud be for the agent at the point of origination to warn the passenger that if they are unable to aquire proper documentation before the international leg, they will be denied bordering, the remaining portion of their reservation cancelled, and the passenger, not the airline, will be responsible for further travel back to their home.