While the effect above just considers mixing due to molecular diffusion, the atmosphere of course also vigorously convects due to the effects of thermal heating (and humidity differences). Both also turn over the atmosphere and mix it.
That said, the composition of the atmosphere does change with altitude. See, for example, here. The thing is that the thickness of layer corresponding to the blurry interface between bromine and air is about as thick as the entire atmosphere, and therefore hard to distinguish from other effects (such as ionization) that change its composition with altitude.