As 3D printing and other advanced manufacturing techniques have become more common, it is increasingly possible to produce structures with nearly arbitrary internal geometric and compositional features, opening up vast new design space for engineers. In this work, we consider a kirigami-inspired, flexible architected material comprising rotating squares joined at their vertices. The rotational degrees of freedom result in significant structural compliance due to the large internal rotations of the squares. While the static properties of these structures (such as their auxetic characteristics) have been studied extensively, much less work has been done on their dynamic properties, especially nonlinear dynamic properties induced by large movement of internal components. Here, we examine the nonlinear static and dynamic responses of these systems, including the propagation of vector solitons and transition waves, and collisions of these nonlinear waves. Finally, we discuss how stimuli-responsive materials can be integrated with the nonlinearities of kirigami-inspired architected materials to enable autonomous behaviors, including movement and control of trajectory, as well as mechanical computing, which enables “information processing” to be treated as a material property.