Based on your observations, I highly recommend having your child evaluated. Initially, a pediatrician should run routine blood tests, urinalysis, and a chest X-ray PA & LAT and if all are OK, then I would consider a pulmonary functions test. Sometimes, bronchial asthma symptoms may develop from exertion, especially in cold weather.
A clue to a potentially cardiac issue of congenital nature should be entertained. An element of Cyanosis on exertion, “bluish discoloration of lips and finger,” may or may not be seen; certain abnormalities on a chest X-ray may provide a clue. An echocardiogram with a pediatric cardiologist may be necessary for a definitive diagnosis.