I am often asked about stretching routines for children. I taught karate to children for many years, and found some stretches needed to wait until the children had a foundation built with their flexibility and balance. This was not an age factor, but an experience factor at work. An eight year old who had trained with me for two years was more experienced than a nine year old who just started classes.
For starters, I recommend a routine for children that involves only seated stretches. We don't want to bring in the balance component too early. Otherwise, the attention for detail (straight back, feet parallel while standing, for example) get sacrificed in the effort to twist, bend, etc., while standing. If you look at the stretches on howtostretch.com the first few pages have seated stretches for a reason. It is really most beneficial to just work on a muscle at first without having to worry about falling!
Next, it is most important to help children with their posture first, and then let them go down as far as they can in a given stretch. When I taught kids, there was always a little friendly competition between the kids to see who could "go down further" in a stretch. While they tried to go lower than each other in the splits, for example, I would remind them to lock their knees and pull back their toes (see splits).
I would reinforce the ideas that would prevent injuries to their joints. The most common mistake I found children would make is best shown in this picture on top, the extended leg is bent. The correct way is below, with the leg straight and the toes pulled back. If you are working with your kids, this is a good place to start.