The effective theory for the dynamics of hot non-Abelian gauge fields with spatial momenta of order of the magnetic screening scale g2T is described by a Boltzmann equation. The dynamical content of this theory is explored. There are three relevant frequency scales, gT, g2T and g4T, associated with plasmon oscillations, multipole fluctuations of the charged particle distribution, and with the nonperturbative gauge field dynamics, respectively. The frequency scale gT is integrated out. The result is a local Langevin-type equation. It is valid to leading order in g and to all orders in log(1/g), and it does not suffer from the hard thermal loop divergences of classical thermal Yang-Mills theory. We then derive the corresponding Fokker-Planck equation, which is shown to generate an equilibrium distribution corresponding to 3-dimensional Yang-Mills theory plus a Gaussian free field.