We study the convergence of distributions on finite paths of weighted digraphs, namely the family of Boltzmann distributions and the sequence of uniform distributions. Targeting applications to the convergence of distributions on paths, we revisit some known results from reducible nonnegative matrix theory and obtain new ones, with a systematic use of tools from analytic combinatorics. In several fields of mathematics, computer science and system theory, including concurreny theory, one frequently faces non strongly connected weighted digraphs encoding the elements of combinatorial structures of interest; this motivates our study.