We consider a prototypical discrete model of phase transitions. The model consists of a chain of particles, each interacting with its nearest and next-to-nearest neighbors. The long-range interaction between next-tonearest neighbors is assumed to be harmonic, while the nearest-neighbor interactions are nonlinear and bistable. We consider overdamped dynamics of the chain and after suitable rescaling obtain a discrete reactiondiffusion equation with a negative diffusion coefficient. Using a

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... ratic nearest-neighbor interaction potential and introducing new variables, we construct and study traveling-wave-like solutions that describe dynamics of phase mixtures in the lattice. Depending on the value of the applied force, phase mixtures either get trapped in one of the multiple equilibrium states or propagate through the chain at a constant speed. At low velocities near the depinning threshold, the motion is of stick-slip type. Numerical results for smoother potentials also suggest the existence and stability of the steady motion in a certain range of applied loads.