  We introduce an extension of the non-equilibrium dynamical mean field theory
to incorporate the effects of static random disorder in the dynamics of a
many-particle system by integrating out different disorder configurations
resulting in an effective time-dependent density-density interaction. We use
this method to study the non-equilibrium transient dynamics of a system
described by the Fermi Anderson-Hubbard model following an interaction and
disorder quench. The method recovers the solution of the disorder-free case for
which the system exhibits qualitatively distinct dynamical behaviors in the
weak-coupling (prethermalization) and strong-coupling regimes
(collapse-and-revival oscillations). However, we find that weak random disorder
promotes thermalization. In the weak coupling regime, the jump in the
quasiparticle weight in the prethermal regime is suppressed by random disorder
while in the strong-coupling regime, random disorder reduces the amplitude of
the quasiparticle weight oscillations. These results highlight the importance
of disorder in the dynamics of realistic many-particle systems.
