I hope you mean by "you push out the old air and pull in new air when you water," that it's only when the water dries from the soil that fresh air is pulled in because that's the only time it happens. The way your write it, this sentence doesn't appear accurate.
From Jeff Lowenfels "Teaming With Microbes", p. 32 (an excellent read by the way):
Water moves between soil pore spaces in one of two ways: by pull of gravity or by the pull of individual water molecules on each other, or capillary action. Gravitational water moves freely through soils. Picture water being poured into a jar of gravel: gravity pulls the water to the bottom as the jar fills up. Large pores promote the flow of gravitational water. As the water fills the pores, it displaces and pushes out the air in front of it. When the water flows through, it allows a new supply of air to move in. When gravitational water hits roots, which act like sponges, it is absorbed.
Smaller soil pore spaces contain a film of capillary water that is not influenced by gravity and is actually left behind after gravitational water passes through. The liquid is bonded together by the attraction of its molecules for each other (a force known as cohesion, but let's not complicate things) and to surrounding soil surfaces (a force known as adhesion). This creates a surface tension, causing the water to form a thick film on the particle surfaces. Capillary water can "flow" uphill. It is available to plant roots after gravitational water has passed by and as such is a major source of water for plants.
Hydroscopic water is a thinner film of water, only a few molecules thick, which, like capillary water, is attached to extremely small soil particles by virtue of electrical properties. This film is so thin that the bonds between water molecules and soil particles are concentrated and extremely hard to break. Roots cannot absorb it, therefore, but this film of water is critical to the ability of many microbes to live and travel. Even when soil conditions are dry, the soil particulate surface holds some hydroscopic water, it is impossible to remove it from soil without applying lots of heat and actually boiling it off.
Just about half the pore spaces in good soil are filled with water. The other half are filled with air.
Water movement pushes stale air out and sucks in air from the surface, so adding water means an exchange of air occurs, which is important. If a healthy soil food web is in place, the metabolic activity of soil organisms uses the oxygen and creates carbon dioxide. The presence of carbon dioxide is a good sign hat the soil contains life; however, the carbon dioxide must be exchanged with fresh air to keep life going.
In some soils, the pore spaces are cut off in lots of places, and air is not exchanged when water flows. In fact, water may not flow at all. These soils have very poor porosity - that is, they lack adequate space between the soil particles. All the oxygen in the soil can be used up by aerobic metabolic activities, resulting in oxygen-less, anaerobic conditions. Organisms that can live in such conditions often produce alcohols and other substances that kill plant root cells.