I love Reverse Robins Au but I feel like a lot of people interpret it as reverse roles Au rather than how I interpret it Reverse Age au.
Damian:
Is dropped off with Bruce at age 8 - violent assassin child who meets tired goth Bruce who's not quite used to being batman yet and is nowhere near equipped enough to raise a child but he's damned if he's not gonna try his hardest.
It's a bit of a mission at first trying to curb the violent tendencies and raise a whole antisocial person (and wow, he admires Alfred so much more now). He almost quits Batman for a bit until he figures out that making Damian his sidekick might be a good outlet for him.
Duke
Bruce is out on patrol with Damian when he finds a little boy who's parents are out of commission and is too stubborn to believe that the Bat would actually help him.
Damian hates that he's not an only child anymore but eventually warms up to the idea when he realises they can team up against Bruce and that they are not in competition.
When Bruce finds out that Duke is a meta he wants to flinch back and send him away, but he doesn't, partially because he loves Duke so much the idea of losing him so quickly repulsed him and partially because Damian pulled a sword on him at the idea.
Tim:
Roughly 9 years after Damian is taken in Bruce, Damian and Duke are approached by a small boy saying he knows who they are and wants to join them. Bruce refuses but Damian spots the opportunity to have someone else be Batman's sidekick whilst he can become a hero in his own right.
Damian starts following Tim around and looking out for him and then he realises that Tim is being neglected and come on he can't just leave Tim there so he brings him home and Bruce can't really argue against that.
Two months later Batman has a new sidekick and Gotham has a new lone hero.
Cass
They only notice her because she wants them to and absolutely no one questions it when she's bought home. All three boys had wanted a sister at some point and none of them cared about her past or her speech.
Jason:
Tim's a teenager now has almost finished highschool when Bruce goes out one night with the batmobile and comes back to where he parked it find a kid stealing the tires. Bruce can sense that Tim is going to inevitably leave him as well so what better time to take in a new kid when he also will need a sidekick soon.
The decision hurts Tim more than anything else because he's being replaced and he's not ready to be on his own even though he's never been more ready.
Jason grows quicker than either of his other two boys and he's more determined than either of them to right the wrongs of Gotham. That determination was what led him to Ethiopia where his the joker killed him.
Dick
Bruce was at the circus at Alfred's suggestion, something to distract him from the empty seat at the table where his youngest should be. It's at the circus where he sees the trapeze line fail and he can't stop himself from reaching out to the falling artists. It hurts him to take in another child so soon after losing Jason but the thought of that tiny boy who saw his parents die going into Juvie almost kills him.
Dick brings joy into the manor that hadn't been known since Bruce was a boy and everyone in the family falls in love with him.
When the Red hood emerges, angry at his older siblings for not protecting him and wanting to kill Dick they're all a worrying amount of relieved and angry. Surprisingly it's Dick that manages to get him to come back to the family.