The dynamic interactions between complex molecular structures underlie a wide range of sophisticated behaviors in biological systems. In building artificial molecular machines out of DNA, an outstanding challenge is to develop mechanisms that can control the kinetics of interacting DNA nanostructures and that can compose the interactions together to carry out system-level functions. Here we show a mechanism of DNA tile displacement that follows the principles of toehold binding and branch migration similar to DNA strand displacement, but occurs at a larger scale between interacting DNA origami structures. Utilizing this mechanism, we show controlled reaction kinetics over five orders of magnitude and programmed cascades of reactions in multi-structure systems. Furthermore, we demonstrate the generality of tile displacement for occurring at any location in an array in any order, illustrated as a tic-tac-toe game. Our results suggest that tile displacement is a simple-yet-powerful mechanism that opens up the possibility for complex structural components in artificial molecular machines to undergo information-based reconfiguration in response to their environments.