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This paper investigates the consensus problem of high-order multi-agent systems (MASs) under multiple communication constraints, including asynchronous communication, arbitrary switching topologies and time-varying delays. In the scenario of asynchronous communication, each agent only receives its neighbors' states information at the time instants when it needs to adjust the state. For each agent, an distributed control protocol involving its own all-order state information and its neighbors' first-order delayed state information is designed. With the help of state vector transformation technology, the high-order consensus problem under multiple communication constraints can be equivalently transformed into a convergence problem of products of nonnegative matrices. The related knowledge of nonnegative matrix theory is used to solve this convergence problem, and further establish a sufficient algebraic condition of achieving high-order consensus. Finally, the achievement of high-order consensus is demonstrated through a numerical example.