I know that Talia has gotten a bad rap in some runs and she isn't always hands on with Damian once he starts living by Bruce's codes but that's stupid and I think that in fact, Talia is hands on though she mothers from afar.
There's a weekly video call, sometimes with Bruce too especially if Damian needs to be praised or needs a taking to. Talia sends gifts and calls him whenever she's a got a break from League of Assassins business. Damian gets in trouble at school and when Bruce can't be contacted, she will take that all over her airpods while in a shoot out with Deathstroke or some shit and it's sure as sunrise that Bruce gets told exactly what went down and her recommended sanctions for that behaviour.
Talia comes to Gotham when she can and Damian gets absolutely spoiled rotten but Talia is also aware that his sibling bonds should be fostered and by singling out her son, it could isolate him. So she has quality time with the Bats too.
Jason, she knows and loves like a son (that part of Lost Days never happened it's icky) and they have a good mother-son relationship. She definitely brings him cool weapons or gives him access codes to League equipment and tells Bruce to go easy on him.
Tim is wary because, I mean, the way him and Damian started out... But Tim of course has League approval and Talia has great respect for him. She will spend hours sparring with him or giving him puzzles to figure out. She counsels Damian to be good to this brother, he may be useful in later life but she's just fond of Tim because he's very much an Al Ghul even if he isn't really.
Cass and Steph love Talia and the three often spend an afternoon doing girly stuff, coffee and shopping. Cass and Talia understand one another and Talia sympathises with Steph because she has a father like that too.
Barbara, of course, is wary. This is Talia al Ghul, known criminal and high profile target on most Interpol lists but Talia wins her over mainly by offering to hook Babs up with elusive contacts overseas.
Duke, fairly new to the family, is just getting used to the fact that the Damian Wayne's mom is the Talia al Ghul so he's very nervous but respectful of Talia. It's 'yes, ma'am' to everything she says or asks and Talia thinks that's sweet, telling Damian to hold this elder brother up as an example of gentlemanly behaviour. She also tells him that if he ever wants other work, in the League, she will hook him up to which Bruce just grunts warningly at.
Dick, oh Richard Grayson. That's a tough cookie. She knew him when he was younger than Damian and he was suspicious of her even then and time has not endeared him to her. However, Talia respects him for all he's done for Damian and the level of care he's given her son and her Beloved. They have mutual grudgingly respect.
The one member of the extended family she doesn't get on with. Selina. Selina may disappear from time to time or become estranged from Bruce but she's fond of his kids and the idea of that... Woman winning them over? Not a chance. Selina always appears with more gifts or her spare time to take them on missions. The two women are cordial. The kids don't pay kind to this since they're getting all this attention and cool stuff and Bruce is more easily worn down when Talia or Selina are on their side. All except Dick because he's been there from the start, he's done this tango and while he doesn't trust either woman with his dad and his siblings, he's the one who calls Selina and tells her Talia is around because he won't let either of them win that easy.
