We study the containment control problem for high-order heterogeneous nonlinear multi-agent systems under distributed event-triggered schemes. To achieve the containment control objective and reduce communication consumption among agents, a distributed event-triggered control scheme is proposed by applying the backstepping method, Lyapunov functional approach, and neural networks. Then, the results are extended to the self-triggered control case to avoid continuous monitoring of state errors. The developed protocols and triggered rules ensure that the output for each follower converges to the convex hull spanned by multi-leader signals within a bounded error. In addition, no agent exhibits Zeno behavior. Two numerical simulations are finally presented to verify the correctness of the obtained results.