Your child coughing after a bout of running could be an early sign of exercise induced asthma. Doesn't mean he has asthma though and also don't stop him from running. Avoid running outside in cold weather as cold air is a trigger.
Next Steps
get him tested for asthma symptoms with your pediatrician.
Health Tips
avoid exposure to cold air, smoke or any other aerosolised irritant.
exercise should not stop, but avoid over exertion.