Adapting to dynamic data distributions is a practical yet challenging task. One effective strategy is to use a model ensemble, which leverages the diverse expertise of different models to transfer knowledge to evolving data distributions. However, this approach faces difficulties when the dynamic test distribution is available only in small batches and without access to the original source data. To address the challenge of adapting to dynamic distributions in such practical settings, we propose Continual Multi-source Adaptation to Dynamic Distributions (CONTRAST) that handles multiple source models and optimally combines them to adapt to the dynamic test data. CONTRAST has two distinguishing features. First, it efficiently computes the optimal combination weights to combine the source models to adapt to the test data distribution continuously as a function of time. Second, it identifies which of the source model parameters to update so that only the model which is most correlated to the target data is adapted, leaving the less correlated ones untouched; this mitigates the issue of ``forgetting" the source model parameters by focusing only on the source model that exhibits the strongest correlation with the test batch distribution. Experiments on diverse datasets demonstrate that the combination of multiple source models does at least as well as the best source (with hindsight knowledge), and performance does not degrade as the test data distribution changes over time (robust to forgetting).