As altitude increases, air pressure decreases. In order for your lungs to fill with air, the air pressure in your lungs has to be less than the pressure of the air outside your lungs. Air moves from areas of higher pressure to lower pressure. This decreases the pressure in your chest and air flows in. When the diaphragm is up, it puts pressure on the chest cavity and the pressure in the lungs is greater than outside the lungs.
Air flows out.
This is an example of Boyle's Law. The movement of the diaphragm is controlled by the brainstem. Anyway-the reason that it is harder for some people to breathe at higher altitudes is that the air pressure differences aren't as great between the inside of the lungs and outside. Argonne National Laboratory, Division of Educational Programs, Harold Myron, Ph.D., Division Director.