We consider dynamic versions of the nonlinear minimum cost multicommodity flow problem and the maximum multicommodity flow problem on networks. These problems generalize classical static flow problems, extend some dynamic problems from ([1,2,3,4,5]). We study the dynamic multicommodity flow problems on networks with time-varying capacities of edges and transit times of edges that depend on sort of commodity entering them, what means that the transit time functions on the set of edges for different commodities can be different. For the minimum cost multicommodity dynamic flow problem we assume that cost functions, defined on edges, are nonlinear and depend on time and flow, and the demand function also depends on time. To solve the considered dynamic multicommodity flow problems, we propose algorithms based on their reduction to static ones on an auxiliary time-expanded network ([3]).