Bistability has become a common strategy in compliant mechanisms, soft robotics, and soft mechanical metamaterials to overcome the intrinsic limitations of soft materials (e.g., high dissipation, low speeds and forces) as well as to create new functionalities (e.g., energy absorption, motion, deployable structures). The responses of bistable structures are determined by their geometric design and the mechanical properties of their constituent material. Here we introduce a strategy to dynamically program the responses of bistable structures by tuning their elastic energy landscape with external magnetic fields. We demonstrate that the stable states of a slender beam, fabricated with a magnetoactive elastomer, can be dynamically controlled to switch between a monostable and bistable response and the more energetically favorable state of a bistable element can be specified. We create multistable metamaterials by arraying a series of such bistable elements and then harness the tunable response of the multistable metamaterial to control the propagation of topological transition waves. Specifically, we demonstrate the external magnetic-field-based control of the direction, speed, shape, and length of propagation of transition waves; without the need to change any of the physical components (beams and linear springs) of the array. We also show reconfigurable functional devices, such as logical gates, where a single physical system can be controlled to act as different gates with an external magnetic field.