We establish asymptotic properties of M-estimators, defined in terms of a contrast function and observations from a continuous-time locally stationary process. Using the stationary approximation of the sequence, θ-weak dependence and hereditary properties, we give sufficient conditions on the contrast function that ensure consistency and asymptotic normality of the M-estimator. As an example, we obtain consistency and asymptotic normality of a localized least squares estimator for observations from a sequence of time-varying Lévy-driven Ornstein-Uhlenbeck processes. Furthermore, for a sequence of time-varying Lévy-driven state space models, we show consistency of a localized Whittle estimator and an M-estimator that is based on a quasi maximum likelihood contrast. Simulation studies show the applicability of the estimation procedures.