I will first elaborate what wind-gusts are not, for mental clarification:
Thermodynamics as you may have encountered it in a physics course, is a static theory, as opposed to dynamic theories.
More specificly that what is taught at university is usually equilibrium thermodynamics - You calculate the equilibrium state of a system where it will end up, but you have no idea how it will do this.
Ignoring the way of how the system achieves an equilibrium state is fine e.e. for gases that do work in machines, as the microscopic energy exchange timescales are small enough (or you could say equivalently: $dQ$ is very small compared to other forms of work). This way you can describe a dynamic process like adiabatic-work with a static theory, so actually historically engineers were pretty lucky.
The work you've linked to (thermodynamics of diffusion) now will surely use concepts of non-equilibrium thermodynamics: when $dQ$ cannot be any longer neglected, but is still small you can approximate the way to equilibrium with energy fluxes that are gradients of the equilibrium variables $F = c \nabla X$ and with linear evolution $\partial_t X = \nabla F$.
However if we have arbitrary energy and also momentum fluxes, equilibrium thermodynamics can't help you anymore (afaik).
Then 'nature' will do other stuff, stuff that we can understand with the theory of Fluid dynamics, that has the navier-stokes-equations very central to them.
Wind generally is generated by differences in heating, so again we will have fluxes in energy and momentum. So referring to your question, wind is the attempt of energy to equilibrate, but at huge energy differences. At small differences it would simply diffuse.
And Wind gusts are now nonlinear or 'chaotic' solutions to the fluid equations (as @tom pointed out).
After reading some more through the Wikipedia-article on non-equilibrium thermodynamics I see that there is considerable research on how to generalize non-equilibrium phenomena with extremal principles. The extremal principle for equilibrium-thermodynamics is simply the minimization of entropy production, but there is no consensus yet on the rest.
So actually, your question touches really recent theoretical research.