This manuscript develops computationally efficient online learning for multivariate spatiotemporal models. The proposed framework relies on matrix-variate Gaussian distributions, dynamic linear models, and Bayesian predictive stacking to efficiently share information across temporal data shards. The model facilitates effective information propagation over time while seamlessly integrating spatial components within a dynamic framework, building a Markovian dependence structure between datasets at successive time instants. This structure supports flexible, high-dimensional modeling of complex dependence patterns, as commonly found in spatiotemporal phenomena, where computational challenges arise rapidly with increasing dimensions. The proposed approach further manages exact inference through predictive stacking, enhancing robustness and interoperability. Combining sequential and parallel processing of temporal shards, each unit passes assimilated information forward and then back-smooths it to improve posterior estimation, incorporating all available information. This framework advances the scalability and adaptability of spatiotemporal modeling, making it suitable for dynamic, multivariate, and data-rich environments. Simulation experiments and an extracted data analysis from the Copernicus Data Space Ecosystem (CDSE) help evaluate and illustrate the framework.