Here is my very basic setup for shooting kids and dogs. Obviously, I bring more for weddings but for kids and dogs, I just bring two Canon 5D Mark iii cameras, the Canon 35mm f/1.4 and Canon 50mm f/1.2 lenses, a Holdfast MoneyMaker (although sometimes I do not use this for kids, more for weddings), hand puppet and some treats for dogs. This all fits neatly into a LowePro Backpack.
I like to bring a minimal setup for family shoots: no lights, nothing complicated. It’s important for me to move fast and think fast and keep it simple. And too much gear, especially lights and reflectors, can be a little daunting and can kill the spontaneity. People freeze up. When I show up the kids don’t see me as a photographer. They see me as someone who came over to play who happens to have cameras. And a green puppet. My shoots are chaotic and unpredictable and I love that.