A new mechanism of thermo-acoustic destabilization of combustors powered by burning droplets is investigated. Acoustic induced heat release disturbances are found to be driven by oscillations of droplet lifetime during the final stage of the droplet combustion before their extinction. The mechanism is the following. Synchronized acoustic disturbances alter the droplet evaporation rate and modify their diameter. Perturbations of the droplet diameter along the droplet trajectories lead to fluctuations of their lifetime. These fluctuations trigger in turn an oscillating motion of the burning droplet cloud boundary that is synchronized by the acoustic excitation. This leads to large heat release fluctuations before droplet extinction and constitutes a thermo-acoustic source. This mechanism is found to be particularly important in solid rocket motors in which aluminum droplets released from the propellant burn individually and are quenched as the droplet diameter falls below a critical residue diameter associated to an inert particle. Analytical models are derived in an idealized configuration where acoustic forcing takes place in the transverse direction to the droplet trajectories. Expressions are derived in the frequency space for the droplet diameter oscillations, the motion of the droplet cloud boundary and the resulting heat release disturbances, that take the form of a distributed Flame Describing Function. This model is used to reveal effects of the acoustic pressure level and of the residue diameter on the resulting motion of the combustion volume boundary and corresponding heat release disturbances. It is further extended to consider the peculiar flow in a solid rocket motor and the heat release disturbance model is compared to results from numerical flow simulations of the motor. While highlighted in the context of solid propulsion, the observations made are believed to be more general and the same mechanisms may persist in acoustically perturbed spray flames from hydrocarbon fuel droplets.