Here is something that works on my iPhone SE with iOS 10.x:

(1) Open the AA App.
(2) Click on the trip you want to update.
(3) Put your finger near the top of the card displaying your first flight.
(4) Drag your finger (without taking it off the screen) UPWARDS as far as you can. The card will move up with your finger.
(5) Now drag your finger DOWN the screen towards the bottom. The card will move down with your finger.
(6) As your finger pass the middle of the screen, the circular arrow-head update icon will show-up.
(7) Keep dragging your finger until the update icon start to rotate (like the old version's update).
(8) Take off your finger off the screen.
(9) Your trip will be updated.

It took me a while to figure this out. I think this is a bug. AA probably will fix it in the next version.

Tintin