Most of the time when I have flown Delta or American, flight attendants have a list of people who have connecting flights and if they get the gate numbers they will announce them - like "Passengers connecting to these cities - Orlando, Gate D4 ; Boston, Gate C2" and so forth. Obviously (and they will state this) you'll need to double check when you disembark in case that has changed.
And often if I'm on a flight where there are tight connections because of a delay, they will encourage (though there is no way to mandate) other passengers to let those connecting to whatever city disembark first. Usually people comply, but there are always some special snowflakes who are not connecting who insist on getting off - usually the ones whose luggage barely fit in the overhead bin.