Вы "ждете" страницы, чтобы "выскочить"
Future<void> pushScreenAndDoSomething(BuildContext context)async{
await Navigator.of(context).pushNamed('YourPage');
print('This executes after the screen is destroyed');
//you can also get a result
String p = await Navigator.of(context).pushNamed('YourPage');
//Inside your page, you Navigator.of(context).pop('DATA');
print(p); //it should print DATA
}