What framework/API to use to capture screen of a device connected to the Mac in the way OBS or QuickTime Player does when an iOS device is connected to Mac via USB.
When you select New Movie Recording in the QuickTime Player you can choose an Connected iPad or iPhone to record it’s screan. Same with OBS.

What is the way to do it in my own MacOS app?

I tried to list devices with AVFoundation and ScreenCaptureKit but only Mac camera, mic and displays are listed. Expected to see an API to list external sources too.