We study a microscopic mean-field model for the dynamics of the electron glass near a local equilibrium state. Phonon-induced tunneling processes are responsible for generating transitions between localized electronic sites, which eventually lead to the thermalization of the system. We find that the decay of an excited state to a locally stable state is far from being exponential in time and does not have a characteristic time scale. Working in a mean-field approximation, we write rate equations for the average occupation numbers ⟨ni⟩ and describe the return to the locally stable state by using the eigenvalues of a rate matrix A describing the linearized time evolution of the occupation numbers. By analyzing the probability distribution P(λ) of the eigenvalues of A, we find that, under certain physically reasonable assumptions, it takes the form P(λ)∼1/∣λ∣, leading naturally to a logarithmic decay in time. While our derivation of the matrix A is specific for the chosen model, we expect that other glassy systems, with different microscopic characteristics, will be described by random rate matrices belonging to the same universality class of A. Namely, the rate matrix has elements with a very broad distribution, as in the case of exponentials of a variable with nearly uniform distribution.