As I mentioned before, leave all the unnecessary items in check luggage. Carry only the umbrella type stroller that they can swift through easily and most likely will fit in the overhead bins (other types are too big and usually have to be gate-checked). Bring back-packs so each adult can carry one on their back and one adult carry the baby and the other carry the stroller.

I'm an FA and from what I see on a day to day basis, most family with infants tend to leave the a/c last because they have too much stuff to carry and other people can't wait for them to go out first. So in the end the only group that usually got left behind is this group (together with those requiring wheelchairs).

2nd of all, ask for the most forward seats so they don't have to wait too long to deplane (this will help if they have some sort of status with UA).