The Model-free Prediction Principle of Politis (2015) has been successfully applied to general regression problems, as well as problems involving stationary time series. However, with long time series, e.g. annual temperature measurements spanning over 100 years, it may be unrealistic to assume stationarity throughout the span of the dataset. In the paper at hand, we show how Model-free Prediction can be applied to handle time series that are only locally stationary, i.e., they can be assumed to be as stationary only over short time-windows. Both one step-ahead point predictors and prediction intervals are constructed, and the performance of model-free is compared to model-based prediction using models that incorporate a trend and/or heteroscedasticity.We also demonstrate the application of our prediction methods to speleothem climate data which exhibits local stationarity and show that our best model-free point prediction results outperform that obtained with the RAMPFIT algorithm previously used for analysis of this data.