I think you touched upon something important here. It is all about the power. Especially if you are a child and you lack any sort of power in your life, you're going to look for it in any small or subtle way that you can.

When I was in elementary (and up to middle school) I was bullied. A lot. And if I wasn't being bullied, I was invisible. At home, my parents doted over my younger brother, spoiling him with attention because he had social problems and learning disabilities (his actual issues weren't diagnosed properly until years later when he was about 12). My dad was controlling and mentally and emotionally abusive. Otherwise, I can't say that my parents are bad people or that I had a bad childhood.

And yet, I had no power in my life. It makes a lot of sense now to me. I've hated myself for a long time for being an abusive, nasty older sister to my brother. He was the only one in my life weaker than I. There weren't any serious injuries, or even bruises, but rather pushing, yelling, making fun of him, etc. It may not seem like much, but that can still have a devastating effect on someone so young who looked up to me. When I was around 10 or 11, I stopped hitting him and went to the teenager phase of "I'm going to ignore you" until I was about 14. It wasn't pretty.

Good news though is that when I saw the error of my ways, my brother and I ended up becoming best friends.

When it's a child bully (whether with other children or with animals) we shouldn't cast that child aside as just a bad child. That child most likely has issues with power, and maybe things will turn out differently for that kid if someone can get down to their level and say "I'm listening to you."