We study the kinetic properties of strongly interacting electron-phonon systems within the validity of Migdal-Eliashberg theory (MET). Going beyond the gradient expansion and the quantum Boltzmann equation (QBE), we solve the time dependent Schwinger-Dyson-Keldysh equations explicitly, and show that the system remains locally stable at arbitrary interaction strength, in contrast to recent claims based on the QBE approximation. Our results prove that strong coupling electron-phonon instabilities necessarily originate from finite bandwidth effects, and that within MET the superconducting $T_c$ is not bounded from above, paving the way for strong coupling electron-phonon superconductivity.