A wall that actually allowed free air exchange from inside to outside and vice versa would be a building physics disaster. With the air, heat and water vapor would also eagerly switch sides, resulting in energy waste and moisture problems up to mold infestation.
Perhaps a brief excursus on where the "breathable" walls come from (text Wikipedia, it’s easier than writing it yourself): "The hygienist Max von Pettenkofer found in 1858 during air exchange measurements in his office that after supposedly sealing all joints, the air exchange rate decreased less than expected and explained this by a significant air exchange through the brick walls. According to current knowledge, however, he had overlooked that rooms have other components than walls and that the stove in his test room was not sealed."
Even in masonry known to be well "breathable," water vapor diffusion – not air exchange – accounts for only max. 2 - 3% of the moisture removed by normal ventilation. When using the description of "breathable walls," the impression is often mistakenly given that there is an exchange of air from inside to outside through the walls. But that does not exist!