How does each airline handle seat assignments for traveling with minors
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AA:AA has an automated system that runs within 15 minutes after a reservation is ticketed, and periodically every day. Keep in mind that it only works if you or your TA haven't selected any seats (you can ask them to remove the seat assignments, and it will assign you seats the next time it runs) and obviously you have no control over which seats you get. They will more likely give you one row across aisles, than two rows behind each other.
United:United allows to choose any seat yourself, including preferred seats if there aren't enough regular seats nearby