I believe the rules changed last season, red cards apply to the next competitive game whatever the competition (unless the red was received in the EFL Trophy where it only applies to the next EFL Trophy game).....accumulated yellows are competition specific though.
It applies to the next game(s) that are considered to be the same or the levels above that which the sanction was awarded. Therefore a red in the EFL Trophy would also attract a ban from the next League game but not vice versa.