An extended stochastic block model for clustering dynamic weighted network data is introduced, where the block memberships are represented by a sequence of latent variables following a Markov chain. Motivated by the availability of original data on patient transfers within a network of Italian hospitals, we propose to rely on a bivariate Poisson to explicitly model the distribution of the dyads conditional on the states occupied by both nodes involved in the relation at a given time occasion. In order to enhance the interpretation of the model parameters, we propose suitable constrains that allow to describe the propensity of nodes to directed interactions. Model inference is based on a variational approach.