Automated Machine Learning (AutoML) systems have been shown to efficiently build good models for new datasets. However, it is often not clear how well they can adapt when the data evolves over time. The main goal of this study is to understand the effect of concept drift on the performance of AutoML methods, and which adaptation strategies can be employed to make them more robust to changes in the underlying data. To that end, we propose 6 concept drift adaptation strategies and evaluate their effectiveness on a variety of AutoML approaches for building machine learning pipelines, including Bayesian optimization, genetic programming, and random search with automated stacking. These are evaluated empirically on real-world and synthetic data streams with different types of concept drift. Based on this analysis, we propose ways to develop more sophisticated and robust AutoML techniques.