Here is a more theoretical but practical explanation:
Back in the days when dinosaurs (and I) roamed the planet and before modern tires were compounded using chlorobutyl and halobutyl inner linings there was a real concern about air pressure loss due to the fact that molded rubber was fairly porous and air migrated through the tire casings over time. To complicate matters, as casing rubber temperature increased, the rubber became more elastic and thus air could potentially seep at a higher rate than when cold. Since Nitrogen gas is a relatively larger molecule than oxygen, helium, and hydrogen it was believed that it would not seep as rapidly as the other smaller gas molecules.
<start boring theory>
The physics of pressure change over temperature is represented by the "Combined Gas Law": (P1 * V1) / T1 = (P2 * V2) /T2 and applies to all gasses and gas mixtures. (Yes, we could split hairs over the R-Factor from the Ideal Gas Law but I'm not going there.) Consequently, it makes little if no difference whether you inflate with air, helium, nitrogen or any other gas. Given constant volume, the behavior remains the same as your tire temperature increases regardless of which gas you use for inflation.
</end boring theory>
With the evolution of modern rubber chemistry tires are now constructed with layers of various rubber compounds, belting materials, and adhesives. The first layer applied to a tire building spindle is a thin but extremely dense layer of chlor- and/or halo-butyl rubber that acts as an "impervious" barrier to air migration. Then, depending on the specification, various layers of other rubber compounds and materials are applied until the whole assembly is ready to be thrown into a mold and cured. This significantly improved general air loss from seepage but didn't completely solve the problem because any damage to the inner liner (usually because of mishandling by a dealer or consumer) would reintroduce seepage at the damage points.
The real problems with air loss are twofold. First, low pressure tires flex more creating a higher heat build up especially in the casing sidewalls. When the heat builds up high enough the adhesives used to bond the belting package to the rubber start to decompose, the belt package starts to shift, the casing expands, and, voila, you get a blowout. Secondly, air migration is a problem because it can seep between the plys and create ply separations that will also lead to blowouts.
Just as important, most compressed air contains moisture and oils (from oil misters installed to operate air tools). These are bad actors on tires as hot moisture will hydrolyze various compounds in tires if allowed to migrate while the oil will act as "plasticizer" or softener. (In fact, if you have picked up a tire cut on the outside of the tire that exposes the belting, you should immediately repair or replace it to prevent road/rain water from entering the belt package otherwise you greatly increase the risk of a catastrophic casing failure.) Thus, the valid reason for using nitrogen is that it is fairly cheap, comes compressed in a bottle that is relatively easy to transport to the track, and contains no water vapor or oils that could adversely compromise your tire casing compounds.
How many times have you cracked the pressure on your airline and watched a cone of water vapor blast from the end of the hose? This is why people install dehydrators on the discharge side of their compressors which absorb and/or condense out 99.9% of water vapor and other possible contaminants.
So the bottom line is this: dry, contaminant free gases are best for your tires whether those gases are air, nitrogen, or some other inert combination.
2005 K12S, HID Low-beam, Lazer Hot-Cam, Panniers, Garmin GPS