When it comes to kids running shoes, you want to keep in mind that you want to both encourage healthy running and protect them from injury.
If you're just getting started -- or restarting -- running, I strongly suggest that you take the time and budget the expense of going to a local running shoes store.
I didn't listen to that advice last year when I began running again for the first time in years. I knew I had flat feet, and size 13 feet, but I didn't feel I could afford to spend a lot of money on running shoes. Therefore, I just went to my favorite place to save money -- Wal-Mart.
Now, I still go to Wal-Mart to save money. I think it's great if you know what you're buying. Then you do get great bargains.

However, if you don't know what you're buying, or at least if you're more dependent on quality than you realize, you can get in trouble. That's what happened to me when I just took their cheapest no-name brand of running shoes that fit me.
However, it didn't take long to see they were starting to wear out -- and I don't even put a lot of mileage on my shoes. I prefer to run more intensively, sprinting in intervals rather than run steadily for miles and miles. True, I was running uphill, but so do many runners.
Eventually one of the hooks for the laces broke, and the upper were separating from the sole.
Plus, at some point I began developing pains in the back of my left leg when I ran at a strong pace. I don't know for certain, but I suspect that this happened from using shoes that weren't so good for my feet, or because I was still running in them even as they were falling apart.
So next I took a friend's suggestion to try eBay. Apparently there're a lot of people there who somehow manage to get hold of stocks of big name brand shoes in models that have been discontinued by the manufacturer, and so they sell them for big discounts.
So I spent some money on a shoe model that's no longer being sold even though the brand is a big name. And I got it in my right size -- 13. However, I neglected to check on the width of my feet. When I tried them on, they were too small despite being size 13s.
So I sent them back. The seller refunded me 75% of the price, which means I still lost 25% and the cost of the return postage which was around $10. So I spent a considerable amount of money with nothing to show for it.
Upon reflection, I also realize I was foolish because I didn't buy a shoe made for actual running, just a fashionable athletic shoe for people to wear everywhere except actually running. If it were a good model of high quality running shoe, it wouldn't be discontinued. So I'm glad in retrospective that it didn't fit. They may have made the pain in my left leg worse.
So I finally got a good idea, and found a local store run and staffed by people who are serious runners themselves. The clerk watched my stride and examined the wear on my current shoes. He looked at my arch. He helped fit me with the Brooks Beast, which my feet love, and assured me that they are well-built. I paid a lot more money than I did to Wal-Mart or the eBay dealer, but I'm still using those shoes.
So, unless you're experienced and know your exact shoe size and running needs, go to an expert at your local running shoe store to help you get started on the right food.
Next: Triathlon Running Shoes -- you need special shoes for triathlons.