If a specific mass of air is polluted, will the pollution stick with it? Let us say a car pollutes 1m3 of air. Is that pollution attached to that specific air mass or will it fall to the ground or take its own path through the air, independent of the air it first polluted?

And how far can it travel, typically?
Air is a fluid, as such it tends to mix and concentrations, of anything, tend to disperse into the wider volume. Erik's cup of tea metaphor is pretty accurate, currents of air will pull apart your exemplar 1m3 and disperse the pollutants therein, the heavier particulate, which is mainly soot, water droplets from combustion, and unburned fuel, will full to the ground fairly quickly. Finer particulate and reactive gases like nitrogen and sulfur oxides will react with water to form acids and fall back to earth with the next rain storm. The exception to this general rule of mixing occurs when there is an inversion layer that puts a "lid" on vertical circulation in a particular area causing a build up of pollutants which often results in smog formation.