We consider the minimum cost multicommodity flow problem in dynamic networks with time-varying capacities of arcs and transit times on arcs that depend on the sort of commodity entering them. We assume that cost functions, defined on arcs, are nonlinear and depend on time and flow, and the demand function also depends on time. Moreover, we study the problem in the case when transit time functions depend on time and flow. The modification of the time-expanded network method and new algorithms for solving the considered classes of problems are proposed.