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It's governed by diffusion. There is an equation ( Einstein eq. I think) that says something about the average distance diffused being proportional to sqrt(time) and a diffusion coefficient which depends on what is diffusing and in what medium. the units of the diffusion coefficient are in m^2/sec because of the square root dependence but it's sort of an analog to the speed.

Delta x = Sqrt(6Dt) in 3d

It's based on brownian motion leading to a gaussian distribution of distances travelled

In still air surely so, where people are I guess there's a lot of convection.

I've wondered something similar, in particular about what happens when I open the window of a closed room. How fast do the gas compositions reach equilibrium again?

Haven't found an answer yet even for very simple assumptions, e.g. 1m³ cube room, 0.01m² window.

This is a great question i'd love the answer to as well.

My intuition tells me the speed particles passively would move through the air would be limited roughly by the speed of sound? That feels like a hard upper bound, though?