For directed relations among a set of nodes with a longitudinal structure, we introduce a dynamic stochastic block model where the blocks are represented by a sequence of latent variables following a latent Markov chain. Dyads are explicitly modeled conditional on the states occupied by both nodes involved in the relation. We mainly focus on reciprocity and propose three different parameterizations in which: (i) reciprocity is allowed to depend on the blocks of the nodes in the dyad; (ii) reciprocity is assumed to be constant across blocks; and (iii) reciprocity is ruled out. The assumption of conditional independence between dyads given the latent blocks is retained. Given the complexity of the model, inference on its parameters is based on a variational approach. An approximate likelihood ratio test statistic based on the variational approximation is also proposed. This allows us to formally test for
both the hypothesis of no reciprocity and that of constant reciprocity with respect to the latent blocks. The proposed approach is illustrated via a simulation study and the application to two benchmark datasets in the social network literature.