This paper considers the asymptotic distribution for the horizontal displacement of a random walk in a medium represented by a two-dimensional lattice, whose transitions are to nearest-neighbor sites, are symmetric in the horizontal and vertical directions, and depend on the column currently occupied. On either side of a change-point in the medium, the transition probabilities are assumed to obey an asymptotic density condition. The displacement, when suitably normalized, converges to a diffusion process of oscillating Brownian motion type. Various special cases are discussed.