Upvote:2
The US does not have exit control so she does not need to show her US passport on the way out. For the airline it does (mostly) only matter that she has the right passport (and if needed visa) for her destination.
If asked why she does not have a visa for the US, she (or you if you are there) can show her expired US passport.
Before she comes to the US the next time she will have to get a new US passport.