Like any parent you should be concerned with the type of shoes your children wear for physical activity, and whilst we immediately think of our son's (and now increasingly daughters') football boots quite often fit for purpose, where as specific junior running shoes are overlooked and fashion type trainers are worn instead.
The foot carries the whole body, the impact of each stride travels through the body with stress on the ankles, calves, legs, back and shoulders. In a fast developing body this action can have long term affects and therefore the choice of junior footwear for athletics (or indeed any general physical activity) is of paramount importance; this means junior running, junior high jump, junior triple jump, and junior javelin and so on*.
It is recommended that by the age of 10 you should seek professional opinion from a Specialist Sports provider when considering correct footwear for your kids.
*Whilst once advanced in any particular field discipline specific footwear is also available, at the very least a high quality junior sports shoe is essential.
After the age of ten it is recommended that juniors should wear a good quality, fit for purpose, running/athletics shoe. A specialist shoe store can evaluate biomechanics but do some research in advance; ask around at athletics clubs, the coaches and other parent's will no doubt have had some experience themselves or will be able to point you in the right direction of someone who can. Buy only a good junior fitting sports shoe that meets your child's structural needs. There are different types: neutral and motion control, both offer cushioning and stability but different types of support overall. Ask questions, ask why and check your understanding. Once you have the general idea you can even purchase future footwear online making sure that you can check basic's and then the fit when the shoes arrive.
Be sure to regularly check on the condition of your child's running shoes. Use the same principles which apply to adults running shoes for usage, wear and tear; although children of course will outgrow their junior running shoes also! Make sure that footwear is always laced correctly and not merely slipped on and off as this can alter the level of support. Lock Laces could be a valid option here even. If you follow the suggestions above, you will be meeting the most basic need for your child's sporting safety, health, wellbeing and ultimately performance.