Sometimes, Bruce forgets his kids will literally fight until their dying breath for him; for each other. He forgets that they will risk it all for he, Alfred, themselves, and each other to be safe. He sometimes forgets that his kids are just that, kids.
Sure, Dick is technically a grown man, with Jason following right behind him; and the others just trickle in, but, that doesn't stop them from being kids. Kids who get angry, sad, lonely, scared, and a little closer to death each time.
And when he finally remembers that his kids will fight tooth and nail for their family; he can't help but be both proud, and terrified. His children, many of which have laid nearly dying in his arms; and the few that actually did die, mean so much to him.
That's why he sometimes snaps when Robin gets hurt, that's why he sometimes panic when Red hood goes silent for a few months, that's why his muscles tighten when Red Robin sways on his feet, that's why he wants to strangle the things that make Dick sad; and he's the same with his girls.
They are his, they are his children, they are strong individuals; and he'll be damned if he doesn't start an apocalypse to keep them and Alfred safe. They are a family, some not by blood, but by bonds.