When Bruce married Selina, him and her daughter had already gotten along pretty well, he just didn’t know how his kids would react with her. Kenzie Kyle. Or her “villain/antihero name”, Vanta. Selina Kyle/Catwoman’s daughter.
When Richard/Nightwing met her, he had a bad feeling about her. Thought she was a nasty thief and kept his distance but was civil. Jason was the opposite, the two were practically best friends the first day they met. He basically never left her side, and laughed and laughed with her all day. When Tim met her, he was weary but still warmed up to her, not really going out of his way to be with her but still being nice towards her. Damian hated her. Thought she was useless and worthless, and a dirty villain. He was mostly angry that he wasn’t the special child anymore. Barbara and Stephanie were very warm towards her and Selina, already welcoming the two women into the family like they were blood related. Cass still has yet to interact with her because she’s too busy doing whatever it is that girl does when she’s not on patrol or training.
Everyone knew that Kenzie wasn’t really a thief, they all knew she wasn’t really like her mother, and her mother actually changed her ways for Bruce, the two adults so in love with each other it didn’t take her much convincing to change. Kenzie practically saw Bruce as her dad, she never really had one growing up and he was so willing to step up for her, to be there for her, that she couldn’t help it. So it’s not surprise that Jason and Bruce are the first ones to check on her when she comes home crying. They can finally coax her out of her room to talk to the rest of the family.