We propose a microscopic scheme that allows to describe ac Josephson effect in superconducting junctions in terms of dissipative quantum dynamics of subgap Andreev bound states. This approach is particularly useful for highly transparent junctions at subgap bias voltages in which case a non-trivial combination of Landau-Zener tunneling between Andreev levels and their instability due to quasiparticle escape into continuum may play an important role. In the low bias regime, we evaluate the non-equilibrium current-phase relation of superconducting junctions at arbitrary transmissions and identify a sub-Ohmic phase-dependent dissipative contribution to the current controlled by quasiparticle dynamics near the superconducting gap edge.