Yes a cold draft can cause muscle pain which is normal.
The cold draft can cause your muscles to contract which can tense up your muscles and then lead to muscle pain.
When you get out of the cold draft then the muscle pain will usually go away after the muscles warm up.
So make sure you dress warmly during the cold weather to avoid getting muscle pain from the cold weather or cold draft.
The same goes for cold weather in general when your body gets cold then it can lead to muscle pain and even your arthritis can flare up when your body gets cold.