Can sudden weather change from warm to cold outside cause runny nose/congestion in children?

Yes. : Yes it can. During this cold season it is possible that the symptoms are related to allergies (called allergic rhinitis) or something called non allergic rhinitis. Both conditions have overlapping symptoms with runny nose and swollen nasal passages that lead to nasal congestion and sometimes sneezing. The important thing is to see a doctor to make the diagnosis and begin the correct treatment.

Vasomotor rhinitis: Many people don't realize the important role your nose has in adding moisture to the air we breathe as well as heating that air to a temperature the lungs can tolerate. The transition from a warm house to the cold outside will often irritate the capillaries in the nose & turn on the drainage. They have a name for it-vasomotor rhinitis.