Coflow scheduling models communication requests in parallel computing frameworks where multiple data flows between shared resources need to be completed before computation can continue. In this paper, we introduce Path-based Coflow Scheduling, a generalized problem variant that considers coflows as collections of flows along fixed paths on general network topologies with node capacity restrictions. For this problem, we minimize the coflows' total weighted completion time. We show that flows on paths in the original network can be interpreted as hyperedges in a hypergraph and transform the path-based scheduling problem into an edge scheduling problem on this hypergraph. We present a $(2λ+ 1)$-approximation algorithm when node capacities are set to one, where $λ$ is the maximum number of nodes in a path. For the special case of simultaneous release times for all flows, our result improves to a $(2λ)$-approximation. Furthermore, we generalize the result to arbitrary node constraints and obtain a $(2λΔ+ 1)$- and a $(2λΔ)$-approximation in the case of general and zero release times, where $Δ$ captures the capacity disparity between nodes.