Motivated in part by applications in model selection in statistical genetics and sequential monitoring of financial data, we study an empirical process framework for a class of stopping rules which rely on kernel-weighted averages of past data. We are interested in the asymptotic distribution for time series data and an analysis of the joint influence of the smoothing policy and the alternative defining the deviation from the null model (in-control state). We employ a certain type of local alternative which provides meaningful insights. Our results hold true for short memory processes which satisfy a weak mixing condition. By relying on an empirical process framework we obtain both asymptotic laws for the classical fixed sample design and the sequential monitoring design. As a by-product we establish the asymptotic distribution of the Nadaraya-Watson kernel smoother when the regressors do not get dense as the sample size increases.