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We consider the black hole dynamical evolution in the framework of a Lorentz-violating spacetime endowed with a Schwarzchild-like momentum-dependent metric. Large deviations from the Hawking-Bekenstein predictions are obtained, depending on the values of the Lorentz-violating parameter λ introduced. A nontrivial evolution comes out, following mainly from the existence of a nonvanishing minimum mass: for large Lorentz violations, most of the black hole evaporation takes place in the initial stage, which is then followed by a stationary stage (whose duration depends on the value of λ) where the mass does not change appreciably. Furthermore, for the final stage of evolution, our model predicts a sweet slow death of the black hole, whose "slowness" again depends on λ, in contrast with the violent final explosion predicted by the standard theory. [ABSTRACT FROM AUTHOR]