Aiming at monitoring a time series to detect stationarity as soon as possible, we introduce monitoring procedures based on kernel-weighted sequential Dickey-Fuller (DF) processes, and related stopping times, which may be called weighted Dickey-Fuller control charts. Under rather weak assumptions, (functional) central limit theorems are established under the unit root null hypothesis and local-to-unity alternatives. For gen- eral dependent and heterogeneous innovation sequences the limit processes depend on a nuisance parameter. In this case of practical interest, one can use estimated control limits obtained from the estimated asymptotic law. Another easy-to-use approach is to transform the DF processes to obtain limit laws which are invariant with respect to the nuisance pa- rameter. We provide asymptotic theory for both approaches and compare their statistical behavior in finite samples by simulation.