Does the gravitational pull of the moon affect the composition of Earth's atmosphere?
The atmosphere is continuously mixed by turbulence, so no sorting of its  ordinary components can occur easily and rapidly. Twice a day change, anyway, is far too fast for anything but some tidal effect on pressure and thus on temperature. Ok, these might indirectly slightly modify some thresholds and balance e.g. between states change (water) and some atmospheric chemistry, by a fraction of what day/night change does. I wouldn't call that "change of composition", though.