While it's not common, the more amicable versions of Dick and Bruce parting ways and Dick taking on the Nightwing mantle. The two mutually realize that it's time for Dick to leave the nest and depart on good terms.
In Nightwing Vol 2 #6 after Dick lightheartedly tackles Tim for breaking into his apartment the two patrol in costume together and while Tim talks about how good Dick is and how he'd trade his computer skills for Dick's street smarts in a minute they both discuss each other's strengths and Dick manages to help boost Tim's confidence. Then in Dick's narration boxes he makes clear he considers Tim family:
Hes family. The closest thing to a brother I have. So excuse me if I get a little severe.
In Nightwing #20 Dick makes an excuse to Barbara (as Oracle before she was aware of Tim's secret identity) so Tim can go check on his dad in the wake of an earthquake.
In Vol 2 #23 Dick's muted but understandable panic at realizing Tim made some kind of deal with Lady Shiva to save Connor Hawke's life. This is something he never forgets as he trails Tim years later when Shiva comes to town to complete the deal by fighting Tim to the death.
In Nightwing Vol 2 #68 Tim finally has someone he feels he can talk to about how poorly his dad is reacting to losing most of his money when he goes to visit Dick and Dick helps make Tim feel better about the situation his family is currently in when it becomes clear he's a bit morose.
Nightwing #142 has one of many instances of Dick and Tim bonding as brothers while Tim helps Dick load up the jet for a mission, and then Dick acts like it's obvious Tim's welcome to join him without an invitation, though Tim doesn't pick up on it till Dick asks if he's coming. Dick leaves Batman a note that he "borrowed some things". Like half the bat-cave's arsenal. Along with the plane and Bruce's sidekick. Reading it actually gets Alfred to Facepalm.
In Nightwing Vol 2 #139 there's some heartwarming mixed with Tear Jerker when Dick and Tim have their first real scuffle. Dick tracks Tim to one of Ra's al Ghul's hideouts thinking he's coming to rescue his little brother and instead discovers that Tim is trying to resurrect some of the many people he had lost in the prior months Dick eventually realizes that it's Tim's choice and hugs Tim when he realizes he can't go through with it and collapses.
Tim... You've got nothing to apologize for. Nothing. I let you make the choice for yourself because I knew you'd make the right one.
#153 opens with a number of Dick's friends helping him pack up his apartment so he can move back to Gotham after Bruce's death.
In an attempt to prove he was the greatest Robin, Damian attacked Tim & Jason at opportune moments he set up to prey on them psychologically, and steals trophies from them to mount on his wall. Dick, however, simply hands him one of his escrima sticks & tells him that he doesn't need to try so hard.
"If you haven't noticed, kid, you're already wearing the "R" on your chest."
In one issue of his self-titled series, Dick muses to himself about the differences between himself and Bruce. In the end, he concludes that if Bruce could go back in time and trade becoming Batman just to have his parents back, he would do so in a heartbeat. Dick admits to himself that, while he misses his parents terribly, he wouldn't trade being raised by Bruce for anything in the world.
After Death of the Family, Dick acts like he's over it. People continually offer him support, but he turns them down. Dick decides to take his frustrations out on some thugs. He doesn't exactly hold back, but Damian of all people is there to hold him back. Damian had been watching Dick for the week, and after some encouragement, and a nice little chat about a confiscated katana and video games, which they plan to play together, they part ways. And then Damian died before the game arrived.
Dick consoling Jen on the death of her parents. Also becomes becomes a Tearjerker if you've been following Forever Evil.