Surface-sensitive diffraction techniques are often used to monitor the smoothness of epitaxial thin films during growth, i.e., the propensity for layer-by-layer growth. Interpretation of such data requires an understanding of the relative importance of various factors that mediate smoothness. These include the adsorption-site geometry, the dynamics of atoms during deposition, and possible transient mobility following deposition, as well as thermal diffusion. Here we present a systematic study of the first three factors, emphasizing the interplay between geometry and dynamics. This is achieved by a comparison of several ‘‘low-temperature’’ far-from-equilibrium growth models where adsorption occurs at on-top sites, bridge sites, or threefold or fourfold hollow sites. Film structure is elucidated through determination of the interface width, density of steps and adsorption sites, the kinematic Bragg intensity, and short-range-order parameters. Exact analysis of nonasymptotic properties of these statistical-mechanical models is in general impossible, and so most results presented are from Monte Carlo simulation.