As far as minimals running kids shoes go, you have a few options: ZEMgear, Merrell, New Balance, and Vibram FiveFingers all make shoes that would fit your kids. My kids have used, and liked all of them (they aren't very picky!). I would probably gravitate towards a more minimal sole, which would put the ZEMgear and FiveFingers at the top of the list.

First for all: Yay for getting out and running together! You will have so much fun. As Damien mentioned, you have a few options for children's minimalist running shoes. I will tell you a bit about our experience and maybe that will help you make a decision. We run as a family (the older three run: 11,9, and 7 while the younger two 4, 2 ride in the jogger). We have found that the ZEM shoes and Five Fingers work very nicely for trail running. Road running tends to be a little different. My older children tend to complain more about their ankles and knees hurting when they use these on the road. So, we tried the New Balance Minimus which they love using on the road. The New Balance has a bit more "cushion," although it is not much. The toe box is still nice and wide, and there is no toe drop. We have not used the Merrells for running so I cannot give you any feedback about those. So, if you are running mostly on trails, I would definitely stick with Five Fingers or ZEMS (and this will be your kiddo's preference; I have one that uses Five Fingers and two that like the ZEMS). If you plan to do most of your running on the road then I would recommend the New Balance Minimus. Hope this helps!