Motor proteins that specifically interact with the ends of cytoskeletal filaments can induce filament depolymerization. A phenomenological description of this process is presented. We show that under certain conditions motors dynamically accumulate at the filament ends. We compare simulations of two microscopic models to the phenomenological description. The depolymerization rate can exhibit maxima and dynamic instabilities as a function of the bulk motor density for processive depolymerization. We discuss our results in relation to experimental studies of Kin-13 family motor proteins.