This paper addresses the sampled-data consensus problem in multi-agent systems with non-convex velocity constraints and switching topologies. The main challenge stems from the nonlinear coupling between constraints and topology changes, along with the integral effect of control inputs on both position and velocity in discrete time. A distributed algorithm is proposed using a time-varying scaling factor to convert the constrained system into an equivalent unconstrained one. Conditions based on stochastic matrix properties are derived to ensure a lower bound for the scaling factor. Consensus is achieved if the switching graphs are jointly connected with a directed spanning tree. Finally, simulations are provided to demonstrate the proposed method’s effectiveness.