I suspect it depends. (How's that for waffling?)

I was on a MR going from DFW-LAS-LAX with 2 hours on the ground in LAS. Well, we were late getting out of DFW and I was really worried that I would miss my connection. We pulled up to the gate just as my LAS-LAX flight was scheduled to depart I walked briskly to the departing gate, which was only two gates down. The GA looked up as I approached and asked "Mr Gamecock?" "Yup" She grabbed my BP as she said "Good boy, get on the plane and grab the 1st seat you see." She shut the door on my hind parts and we pushed back. Clearly she waited.

My daughter was on a delayed BA flight from AMS-LHR that was late getting out and had to get from T-5 to T-3. When she arrived at her T-3 gate the AA agent welcomed her and said "we've been waiting for you" and shooed her on board. Her luggage even made it.