Will an air-filled tire normally gravitate toward being nitrogen-filled just by refilling with air?

This probably depends on the tire structure; in general:

• for hydrodynamic size hole there will be no difference in move of oxygen and nitrogen
• for a very small holes there will be some sorting due to effusion, but this is pretty inefficient process since atomic masses of oxygen and nitrogen are similar; what's worse, it will work in the opposite direction -- oxygen molecules are a bit heavier than nitrogen ones, so due to the Graham's law the nitrogen will escape about 1.07 times faster than oxygen.

I thought the big attraction of nitrogen was its low reactivity. The oxygen can chemically combine with rubber molecules, sort of like conbustion, but far below ignition temperature, and the claim is that that effects the tire lifetime. These oxidation reactions will rates will be much faster at higher temperature, but tires do heat up while driving. I was talked into N2, the last time I got new tires. I don't know if it is was really worth the approx $8 per tire it cost me however (i.e. how much the tire lifetime improves and the slight improvement in vehicle milage is actually worth). So if the O2 molecules are diffusing out faster, but more importantly chemically combining with the rubber, I would expect the O2 concentration to decrease with time.

The nitrogen in a tire does not stay in the tire longer than oxygen.

The reason nitrogen is used in high-altitude aircraft tires is that it is a dry, non-reactive, cheap gas; dry being the operative word. Condensation in aircraft tires can freeze and cause small leaks in the tire valves during flight, which can lead to catastrophic landings.

Using a dry gas such as nitrogen in auto tires can result in a more consistent tire pressure, because water condensation and evaporation can change the pressure. This can be important in high performance situations such as race cars.

However, the reason nitrogen is most often used in auto tires is marketing. It's cool because airplanes and race cars use it.

An additional comment on the "benefit" of using nitrogen instead of air in your car tires. Air contains water vapor, the amount of vapor depends on the humidity. For a cold, inner tire surface, the water vapor can condense thereby removing the contribution of the partial pressure of the water vapor to the overall pressure in the tire and lowering the tire pressure. Nitrogen is non-condensable at ordinary conditions. So, nitrogen instead of air could keep the tire pressure slightly higher when the tire is cold. Once you start driving, friction warms the tire and the condensed water evaporates; so overall the claim that nitrogen is an improvement seems suspicious to me.