The nonsmooth finite-sum minimization is a fundamental problem in machine learning. This article develops a distributed stochastic proximal-gradient algorithm with random reshuffling to solve the finite-sum minimization over time-varying multiagent networks. The objective function is a sum of differentiable convex functions and nonsmooth regularization. Each agent in the network updates local variables by local information exchange and cooperates to seek an optimal solution. We prove that local variable estimates generated by the proposed algorithm achieve consensus and are attracted to a neighborhood of the optimal solution with an O((1/T)+(1/√T)) convergence rate, where T is the total number of iterations. Finally, some comparative simulations are provided to verify the convergence performance of the proposed algorithm.