I think there must be a courtesy rate. During a connection in SFO, we missed our flight to Frankfurt since our plane was delayed coming in. I remember the gate agent going well beyond her job duties walking us over to Lufthansa getting us two Business class seats (only seats left) then walking us through security to bypass the lines so we can board the Lufthansa flight in time. Totally saved our european vacation although we didn’t get our luggage until four days later.
This was United so it really depends on the person, not the airline.
