In 2016, your child can play about any sport year round. Your youngster can literally play more basketball games in a year than Steph Curry and more soccer games than Lionel Messi. Running is no different, you can run indoor track in the winter, move onto spring/summer track and then spend the fall running cross country, not to mention roads races year round.
One of my favorite books of the last five years was David Epstein's Sports Gene, and he has an excellent article in the NY Times, Sports Should be Child's Play. Check it out, keep sports fun and hold off on specializing until your child is older, either late middle school or high school.
One of the problems with youth sports is once adults figured out they could make money off it, sports that were once seasonal become year round. Parents want whats best for their kid and they worry if their son or daughter doesn't compete year around, that they'll fall behind their peers.
The question that begs to be asked, is that good for my kid and in particular their running development?
In short the answer is no and I think with running it's even more important to not specialize early on. Running is not a skilled sport, the way soccer, basketball or volleyball are. This means the learning curve is not as steep, which makes it an easy sports to get started in and to have early success in.
The other advantage of not competing year round is it makes improvement positive linear. If a child races year round, yes they will have an advantage over other 10 year olds, but the disadvantage is it will allow them to reach their peak at a younger age. If your child takes a more moderate, long term approach, it will allow them to continue to make marginal gains as they get older and not run the risk of getting burnt out.
In Sports Gene, David Epstein talks about the two factors that make an elite runner. The first is genetics and the second is adaptability to training. Which essentially means you can have two kids of similar ability, put them through the exact same workouts and three months later one of the kids is beating the other kid by 45 seconds in a 5K. Not everyone responds to training in the same way.
I think it's easier to get burnt out on running than other sports, because running isn't always inherently fun for kids, in the way kicking a ball, catching a football, or shooting a basketball can be. Kids run all the time, but it doesn't have to be in a structured running practice to stay in shape, so let them play.
I've always loved cross country over track, just my personal preference and I competed in both. For young kids, I think your better off putting your kid in active sports like soccer, basketball, lacrosse in the off season where they can still be active, but not feel like it's just running. These sports also develop foot and hand eye coordination that running cannot, as well as lateral movement.
If running is your child's sport, perhaps wait until high school to let them compete in cross country and track and field on a year around basis.