We generalize the quantum transport Landauer-Büttiker multi-channel conductance formula to the case where phase-breaking electron- “phonon” scattering is present. The electron-phonon interaction Hamiltonian is approximated to be that of a local phonon model, which can be solved exactly. This enables one to treat the phase-breaking effect in mesoscopic conductance problems via the knowledge of the electron-phonon coupling constant and the amplitude transmission coefficients for the one electron problem in the presence of a given impurity scattering potential for an energy range h ̷ /τ ϕ around E F. We illustrate the applications of this new transport formalism by an Einstein phonon model, and the suppression of hc/ e oscillations of a ballistic Aharonov-Bohm ring.