You may see a message indicating that the video has been redirected, or the connection to the base unit may appear to have dropped the connection. The splash screen may appear on the base unit monitor.
In fact, the full-screen mode (player or CAST mode), Apple iOS drops the connection to the iOS device and establishes a new connection from the base unit to the video source (YouTube).
The network firewall must enable *.googlevideo.com on port 443 from the base unit bi-directionally.
Attempting to display the IOS device video from within an App will result in the base unit returning to the splash screen.