Simply take input from the TextEditingController.text of the text field, rather CupertinoTextField and make calls with the search query every time the user changes something.
Make sure to display some generic search results from the beginning itself when user input is empty.
Movies/ web series search
keypoints
TextEditingController.text
of the text field, ratherCupertinoTextField
and make calls with the search query every time the user changes something.