Dick takes his role as oldest brother very seriously. It was a little rough in the beginning with Jason but then he got the hang of it and as he got more siblings, he got the hang of it and just completely accepted it and went with it. Obviously he does the constantly having snacks for his siblings, on patrol or just around his house.
I'm pretty sure he has a motorcycle right, so I bet he's picked Damian up from school on it just to give the 'cool older brother impression' to his classmates. He's probably done it for the others when they were younger at some point just as a surprise or because people would probably think twice about bullying them if they get picked up from school on their older brother's motorcycle. Maybe. Even if it doesn't work, it's a nice surprise for whoever's getting picked up.
If he's with Steph or Cass and a guy is being weird with them then he obviously steps up and accesses his inner Batman intimidation game and then laughs with whichever/both girl(s) after about it because "oh my god, he looked like he was gonna shit his pants" and "how long did it take you to master the Batman voice, it was terrifying". Alternately, if one of the girls (usually Steph, maybe even Tim) like a guy, then Dick teams up with Jason to shovel talk/approve of whoever the poor guy is. Either way, he treats his sisters like princesses and makes sure they know how a gentleman should treat them.
Gala's are where he really steps up his game though because there are so many terrible people at them even if he's chatting with someone who's actually nice but he sees one of his siblings in trouble or uncomfortable with another guest, then he will leave and even purposely turn the creep's attention on him if need be.
At home though, he still takes one for the team and Alfred and Bruce know that so basically what ends up happening is that if Alfred's mad at one of them - for like a stupid little reason - then he'll cover for them or even take the blame. Thing is though, Alfred knows that whatever Dick's taking the blame for, he probably wouldn't have actually done and he's just saving someone else so Alfred still has to make Dick face some consequences but lighter than he would have because he knows Dick didn't do it. It also works as a strategy to figure out who actually did it because they usually feel guilty that Dick's getting 'punished' for something they did (unless it's Jason - he doesn't care about Dick - but he still feels guilty because he's lying to Alfred essentially and he respects the man too much for that).