Okay so now is time for the long rant of Dick and Bart's dynamic. I'd like to state rn before anyone @/s me, I don't ship these two romantically, if anything it's more brotherly companionship than anything.
To start off, Bart would definatly be broken up about Wally's death just as much as Dickie would, and in a weird sense they'd have sort of an understanding. After Wally disapears, Artemis goes off grid so it's not like he can talk to her. Barry is off-limits for obvious reasons. (honestly some things Barry's said in the comics makes me feel like he blames Bart for Wally) More often than not, Bart would go to Bludhaven to talk to Dick and find comfort. Dick was, after all, Wally's best friend. And Bart would give him the same comfort.
Dick had already known Bart, he's seen him hang around the mannor, following behind Tim like a puppy with it's tail wagging. For a long time, he'd thought the two were dating with how close they were. Dick, being the awkard brother that's trying his best to get closer to his siblings would try and get to know his friends better and Bart really enjoyed his company whenever Dick stuck around.
Before Wally's death, Bart would always do anything and everything to make him jealous, upset, or a combination of the two. So every time Dick and Wally were in the same room with him, Bart would use his pick up lines on Dick just to piss Wally off and it worked every time. Dick, of course, knew what was happening because Bart had given him a rushed explination before he started doing that and he found it hilarious how Wally's face matched his hair as he dragged Bart away with the younger speedster yelling out "CALL ME". Tim pinched the bridge of his nose when he learned of this and Jason finds it hilarious as well. Damian, not so much.
Bart would definatly give Dick all the hugs, and Dick- being the touch starved boy he was with brothers who aren't too touchy and neither was Wally- always allowed it. He never really did anything in company but whenever they're having silent time just hanging out Bart would lean up against his shoulder or lean back to back with him. Speedsters are also actual furnaces so it's nice in the winter time. Dick would return the affections by a random hair ruffle in public or a touch to the shoulder to show he was there.
Just Batboys and Bart things