there is a huge difference between the behaviour of an encapsulated gas body (like, for example , in a balloon) leading to buoyancy (in an external gas body of different density), and the behaviour of "free" gas particles: don't confuse the physics of the one phenomenon with the other.
whenever two (initially separated) gas bodys are allowed to mix, they'll do so, and, on a macroscopic scale, will result in a homogeneous mixture (entropy rules!) with no difference with respect to composition wherever you went, horizontally.
Vertically, with respect to some tens of kilometers, the composition will change however, following the individual height distribution of each gas in the respective gravity. Hence, if you had an air/hydrogen mixture , there always will be hydrogen even at the bottom , i.e. not all the hydrogen will disappear to the top (in fact, there always will be a higher hydrogen density at the bottom than at the top).
However, the height distribution of hydrogen will reach out much much further into space (as it is a function of the MW of the respective gas), and eventually gravity will become too weak to capture hydrogen forever (on earth, that is: Jupiter would be something completely different, and hence has a hydrogen rich atmosphere)
The same is true for He , and as a consequence, Earth's atmosphere is low on both gases (though they're the most abundant elements in the universe): we're "bleeding" them away (just like Mars did with water, oxygen nitrogen...), and that's why Moon doesn't have any atmosphere worthy to talk of (besides that Earth always would try to steal it)