We study the continuous limit of minimum-cost flow problems on large random graphs. Our main result asserts the convergence of minimum-cost flows to currents minimising a continuous linear-growth functional. Due to the particular growth regime, one of the main features and challenges is the presence of finite-energy flows with singularities. In this talk, we discuss the main ideas and tools to handle this framework, in particular the subadditive ergodic theorem and the blow-up method.