A study of the dynamics of a tunneling particle in a driven bistable potential which is moderately-to-strongly coupled to a bath is presented. Upon restricting the system dynamics to the Hilbert space spanned by the M lowest energy eigenstates of the bare static potential, a set of coupled non-Markovian master equations for the diagonal elements of the reduced density matrix, within the discrete variale representation, is derived. The resulting dynamics
is in good agreement with predictions of ab-initio real-time path integral simulations. Numerous results, analytical as well as numerical, for the quantum
relaxation rate and for the asymptotic populations are presented. Our method is particularly convenient to investigate the case of shallow, time-dependent potential barriers and moderate-to-strong damping, where both a semi-classical and a Redfield-type approach are inappropriate.