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The availability of relational data can offer new insights into the functioning of the economy. Nevertheless, modeling the dynamics in network data with multiple types of relationships is still a challenging issue. Stochastic block models provide a parsimonious and flexible approach to network analysis. We propose a new stochastic block model for multidimensional networks, where layer-specific hidden Markov-chain processes drive the changes in community formation. The changes in the block membership of a node in a given layer may be influenced by its own past membership in other layers. This allows for clustering overlap, clustering decoupling, or more complex relationships between layers, including settings of unidirectional, or bidirectional, non-linear Granger block causality. We address the overparameterization issue of a saturated specification by assuming a Multi-Laplacian prior distribution within a Bayesian framework. Data augmentation and Gibbs sampling are used to make the inference problem more tractable. Through simulations, we show that standard linear models and the pairwise approach are unable to detect block causality in most scenarios. In contrast, our model can recover the true Granger causality structure. As an application to international trade, we show that our model offers a unified framework, encompassing community detection and Gravity equation modeling. We found new evidence of block Granger causality of trade agreements and flows and core-periphery structure in both layers on a large sample of countries.