I think that, in order to do what the Bats do, they have to be pretty competent with kids.
Obviously this stuff is pretty relative depending on the age of the kid and the circumstances, but here goes;
I think Dick Grayson might actually the worst of the Bats at dealing with kids. This isn’t to say he’s bad with them, just that he’s worse than the rest of his family.
He’s A+ at dealing with small kids, like, the really lil ones (he would give them airplane rides all the time), or in teaching situations (like his gym job), but I think that, the second a kid is old enough to detect condescension, Dick’s out. Three-quarters of that is his personality, but kids have a serious problem with being talked down to, and that’s basically Dick’s natural state.
The worst part of this is, naturally, that he thinks he is really great with kids, in spite of the evidence to the contrary.
Jason is the guy that always gets on the kids’ level. He remembers vividly what it was like to be small and not know if you could trust an adult when you couldn’t even look them in the eye, so he always crouches down or sits on the floor to make sure the kid’s comfortable. He’d also unconsciously pitch his voice higher, and he calls little boys, like “lil baller” or “lil dude”, and little girls “darlin’ ” and “lil lady”. He also fist-bumps every little kid in the whole world, even as the Red Hood.
He’s pretty easygoing and is also a sensitive soul, so I think he’d be great at hanging out with kids and making sure they are happy and looked after. (And he wouldn’t mind being a canvas for magic markers, nail polish, or glitter hair clips.)
Plus, he makes them giggle when he does an accidental swear, ie. “Shi–! …vers. Shivers. Yep. That’s what I meant to say.”
Tim talks mostly to kids like they are tiny grown-ups, handshakes inclusive. Naturally, kids love this.
Also, he’s a really patient guy, so he’d be pretty happy to let the kid be mostly in charge of the interaction. He is also full of really random trivia, facts, and knowledge on how stuff works, so any kid in a “why/what does that do/how does this work” phase would follow Tim around with stars in their eyes, because he knows everything and he never tells them to leave him alone or “just because”.
He is also impossibly good at hide-and-seek, and is not above bribery (“incentivising”).
Damian is a wild-card. I actually think he’d be really great with young kids, and is very comfortable dealing with babies and toddlers (except nappies, obviously). He, like Tim, would talk to them like they’re little adults. Except unlike his interaction with adults, he tones down the sarcasm and rude remarks. But not the scowl.
I feel like his aptitude with kids is a consistent source of bewilderment for his family, and if ever one of them asks “How in the heck are you so good at dealing with kids?” he would give them a look of complete disgust, and then continue on as if he hadn’t heard them.
Also, he’d draw pictures for them to colour in.
Babies and kids love Bruce. Which is weird, because he is very uncomfortable dealing with them.
I do think he’s surprisingly good at interacting with kids (possibly because they recognise, in Bruce, a fellow immature baby) but he doesn’t feel like he knows what he’s doing.
I do think he’d be a pretty questionable babysitter, though, especially to younger kids. He can’t incite a conversation with an adult, never mind a kid. If it were up to him, he and the kid would probably sit in complete silence for hours.
I don’t know about Dick, if he was too condescending, he and Damian wouldn’t be so close and have such an effective partnership, since the first thing Damian established upon his arrival is that he would NOT BE PATRONIZED (or he’d break Tim’s face), and he’s not quite as much of an adult as he imagines he is.