It's fine for Tim to have moved on to being Red Robin already, I'm just not sure Damian was officially made the new Robin. That said, with Bruce disappearing I'm pretty sure the only in-character response is for Damian to attempt to help patrol the streets in his own way. Possibly by appointing himself Robin anyways without an official go-ahead from Bruce.

As for his status in the family, Damian might occasionally daydream about going back to the League but Talia and Ra's have basically dumped him there for over a year now, he hasn't taken the opportunity to leave when he's had the chance and even if he started raiding Demon Fang hideouts to find the League or the others kicked him out  now there'd be no guarantee he'd have anywhere to go, so he's part of it. Unless they all feel comfortable with a tiny, homeless assassin with skewed morals wandering the streets unsupervised.
[@Nightwing95]