This paper aims at providing statistical guarantees for a kernel based estimation of time varying parameters driving the dynamic of very generals classes of local stationary processes. We consider coupling arguments in order to extend the results of Dahlhaus {\it et al.} \cite{DRW} to the local stationary version of the infinite memory processes in Doukhan and Wintenberger \cite{DW}. The estimators are computed as localized M-estimators of any contrast satisfying appropriate regularity conditions. We prove the uniform consistency and pointwise asymptotic normality of such kernel based estimators. We apply our results to usual contrasts such as least-square, least absolute value, or quasi-maximum likelihood contrasts. Various local-stationary processes as ARMA, AR$(\infty$), GARCH, ARCH$(\infty)$, ARMA-GARCH, LARCH$(\infty)$, \dots, and integer valued processes are also considered. Numerical experiments demonstrate the efficiency of the estimators on both simulated and real data sets.