They only carry over if it is for violent conduct which by the sounds of it it is. Happened with us with Da Costa the year we went down so it is possible that the rule may have changed since then.
U23 sendings off don't result in suspensions from Championship games ( http://the72.co.uk/74880/leeds-united-red-mist-for-first-team-midfielder-in-u23-game/ ), so I doubt they do for Premiership ones
Straight red after half an hour for an off the ball incident. Finished 0-0 at St James Park Teams play again there on Sat in last 16 of the Premier League Cup ☕️