The significant discrepancy observed between the predicted and experimental switching fields in correlated insulators under a DC electric field far-from-equilibrium necessitates a reevaluation of current microscopic understanding. Here we show that an electron avalanche can occur in the bulk limit of such insulators at arbitrarily small electric field by introducing a generic model of electrons coupled to an inelastic medium of phonons. The quantum avalanche arises by the generation of a ladder of in-gap states, created by a multi-phonon emission process. Hot-phonons in the avalanche trigger a premature and partial collapse of the correlated gap. The phonon spectrum dictates the existence of two-stage versus single-stage switching events which we associate with charge-density-wave and Mott resistive phase transitions, respectively. The behavior of electron and phonon temperatures, as well as the temperature dependence of the threshold fields, demonstrates how a crossover between the thermal and quantum switching scenarios emerges within a unified framework of the quantum avalanche.