Multistable systems, extended in one spatial dimension, can make transitions from one state of local stability to another local state of stability, in a spatially uniform way, or by motion of an interface. The literature on Sine-Gordon and φ4 solitons, in nonlinear dipersive systems, emphasizes their role as phase boundaries of this sort. We emphasize that shock waves can also act as moving phase boundaries in multistable systems. Our analysis invokes ferroelectric transmission lines as the principal model, but also treats linear arrays of coupled bistable springs. The soliton causes a transition in a given local volume, passed by the disturbance, by directly controlling the multistable degree of freedom, whereas the shock wave controls the corresponding force.