We propose new primal-dual decomposition algorithms for solving systems of inclusions involving sums of linearly composed maximally monotone operators. At each iteration, only a subset of the monotone operators needs to be processed, as opposed to all operators as in established methods. Deterministic strategies are used to select the blocks of operators activated at each iteration. In addition, asynchronous implementation is allowed. The first method provides weakly convergent primal and dual sequences under general conditions, while the second is a variant in which strong convergence is guaranteed without additional assumptions. The novelty of this class of algorithms will be discussed and comparisons with the state of the art will be performed.