We present a magnetoelastic lattice in which a localized external magnetic field, generated by an assembly of fixed magnets, tunes the potential landscape to create monostable, bistable, and tristable configurations. Focusing on the tristable potential, we numerically and experimentally confirm the existence of three distinct types of transition waves, each characterized by unique amplitudes and velocities, and establish a scaling law that governs their behavior. We also examine how these transition waves interact with the system's finite boundaries. Furthermore, by adjusting the potential symmetry through the localized external field, we investigate wave collision dynamics. In lattices with asymmetric potentials, the collision of similar transition waves leads to the remote nucleation of a third phase. In symmetric potentials, the collision of dissimilar transition waves results in the formation of a stationary domain wall, with its width tuned by the shape of the tristable potential well.