This paper studies the nonconvex input constrained containment control of multi-agent systems with high-order dynamics and external disturbances. The communication topologies adopted in this study are directed and dynamically switching. A decentralized nonconvex input constrained algorithm with communication delays is adopted. Based on model transformations and convexity analysis, the main results prove that every follower eventually converges to the neighborhood of the leaders’ convex hull, and an upper bound for this neighborhood is derived. Finally, simulations are provided to validate the theoretical findings.