A lot depends on how far you have to go -- whether to an adjacent gate or across the airport. However, my time-saving tips for any connection are:
1 - Check in for both flights ahead of time. Get seats as far toward the front of the plane as you can for your first leg so you can be among the first to disembark -- as long as you're not traveling carryon only. (If they run out of overhead bin space for the last to board and you have to gate check a bag, you potentially might have to pick up your bags during your layover -- not something you want to do with a tight connection!)
2 - Know your connecting gate ahead of time and print out a map of the airport so you know exactly where you need to go once you land. Listen for any gate changes that might be announced before you disembark, and as soon as you disembark, consult a times board to confirm that the gate for your next flight hasn't changed, so you don't waste any time going in the wrong direction.
3 - Before the plane begins its descent on your first leg and that "seatbelt" light goes back on, have the kids visit the restroom so nobody has to go while you're trying to cross an airport terminal (potentially). Similarly, be sure that everybody has put all their "stuff" (electronics, etc.) back in their bags (or in the case of jackets and shoes, put back ON anything that was removed in-flight) well before it's time to disembark. You don't want to have to wait for somebody to tie their sneakers, or run back to find the charging cord they left in their seat.
4 - Travel light! The fewer/lighter bags you have (I would keep it to no more than 1 bag for each child to keep track of, if any -- either a lightweight roller bag or a backpack), the faster you can move. Even if you land early and have plenty of time to make your connection, it's better to "hurry up and wait" at the gate. Delta requires you to check-in (and often begins boarding) 30 minutes or so before the scheduled departure time, and you don't want to miss anything.
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