That's funny, this is something that I discovered about my dog recently. My cousins and their kid, my 4 year old nephew, came to visit me in the US from Holland. They stayed at my 1-bedroom apartment for 5 days. My dog had not been really exposed to children (unless there are children at public places) and I certainly had never had a child at my home before my nephew. I was quite nervous on how he will behave around my nephew, but it went surprisingly well.
I don't know if it was a paternal instinct, or his breed, but whenever the 5 of us are walking (be it at a mall, on the streets, the beach etc.) and my nephew runs and runs away from us (because he's a toddler...), my dog will bark and bark and bark until he comes back. Whenever he sees my nephew fall down, he will bark and bark and bark until one of us helps my nephew or we tell him my nephew is ok. Once, we were at the beach boardwalk and my cousin-in-law was at the beach, my nephew came running to him solo, my dog barked until he saw that my cousin was running after him. I was genuinely and pleasantly surprised by this. My nephew learns to stay with us too and not run away from us because he's been told that 'He (my dog) doesn't like it when you run from us.'
One thing I can tell you is that I trained my dog the 'look' command. Whenever he barks because he thinks my nephew is in distress, I will say, 'Look,' I get an eye contact, he calms down, my nephew comes back. Win-win situation. Based on your post, if it has become an issue for you, you would probably need to step up your game on gaining your dog's attention. Teach him 'look' and 'quiet.' That's what works for me at least.
