When you sleep with your mouth open, does air pass through the nasal cavities? If not, how will you describe the air that enters the mouth?
When you breathe, air follows the path of least resistance. Since the mouth (to pharynx to trachea) has a larger diameter than the nostrils, most of the air will pass through the mouth (unless obstructed by the tongue, etc.)
However, some will enter the nose and proceed towards the lungs by virtue of decreased air pressure in the nasal cavity while inhaling.