We present a model of phase transformation kinetics at the surface of nonvolatile and volatile solids, with special consideration of finite specimen size in one dimension. For nonvolatile materials in which nucleation occurs randomly and homogeneously throughout the sample, a slowing of surface phase change kinetics is predicted as specimen thickness decreases. This deceleration originates from a reduction in the number of grains contributing to surface transformation and the progressively more two-dimensional growth geometry of each grain as the sample becomes thinner. Sublimation increases the relative impingement velocity of subsurface nucleated grains and the interface, thus accelerating surface phase change (versus nonvolatile materials) in the absence of thickness effects. In thin films, the accelerating influence of sublimation competes with the retarding effect of finite specimen size, and thus transformation kinetics can be faster or slower than for nonvolatile materials, depending upon sample thickness. Finally, the model is fit to experimental data for the crystallization of volatile amorphous solid water films of varying thickness. Good agreement between experiment and theory is found, suggesting that our model captures the essential physics of simultaneous surface phase change and sublimation for finite specimens.