In a previous study, the early stages of self-assembly in nanophase materials were explored by coupling a kinetic mean-field analysis with a lattice-based stochastic theory [J. J. Kozak et al., J. Chem. Phys. 126, 154701 (2007)]. Recent experimental results on the postnucleation stages of zeolite assembly and protein crystallite formation have suggested a new study, presented here, in which the docking of a platelet on the existing surface of a structured crystallite is similarly investigated. A model is designed which allows the quantification of factors affecting docking efficiency; principal among these is the structure of the template itself, which here is assumed to be either unstructured or bifurcated into terraces and edges/ledges. Going beyond our earlier study (in which diffusion was restricted to d=2 dimensions), the diffusion space here is enlarged to consider both d=2 and d=3 dimensional flows. By expanding the external diffusion space systematically, we are able to document the consequences (as regards docking efficiency) of diffusive flows in the near neighborhood of a developing crystallite versus surface-only processes. Particularly in regimes where the barriers to surface diffusion are high, and/or the probability of desorption significant, we find that d=3 dimensional processes (leading to a "direct hit") can compete kinetically with surface-only mediated processes. Although the crystallite model studied here is simple, it can be diffeomorphically distorted into a manifold of possible geometries; in analogy with the classical theory of corresponding states, we argue that the familial relationship among these structures suggests that the generic results obtained provide a qualitatively correct description of the kinetics of docking on structured surfaces.