The widespread deployment of wireless and mobile devices results in a proliferation of decentralized spatio-temporal data. Many recent proposals that target deep learning for spatio-temporal prediction assume that all data is available at a central location and suffers from so-called catastrophic forgetting, where previously learned knowledge is entirely forgotten when new data arrives. Such proposals may face data privacy concerns and may experience deteriorating prediction performance when applied in decentralized settings where data streams into the system. To bridge the gap between decentralized training and spatio-temporal prediction on streaming data, we propose a unified federated continuous learning framework, which uses a horizontal federated learning mechanism for protecting data privacy and includes a global replay buffer with synthetic spatio-temporal data generated by the previously learned global model. For each client, we fuse the current training data with synthetic spatio-temporal data using a spatio-temporal mixup mechanism to preserve historical knowledge effectively, thus avoiding catastrophic forgetting. To enable holistic representation preservation, the local models at clients each integrates a general spatio-temporal autoencoder with a spatio-temporal simple siamese network that aims to ensure prediction accuracy and avoid holistic feature loss. Extensive experiments on real data offer insight into the effectiveness of the proposed framework.