We studied structure development in block copolymer films with a regular surface roughness induced by the deposition on corrugated supports. Upon thermal annealing, the residual surface roughness in thin films of cylinder-forming polystyrene-b-polybutadiene (PS-b-PB) di-block copolymer was found to be independent both on the averaged starting film thickness h(0) and on the initial thickness gradient induced by spin-coating. Surface patterns in PS-b-PB films on linear and squared wave gratings appear in a form of spatially registered cylinder and nonbulk lamella phases with the morphological sequences coupled to the topography of the substrate. The observed complex phase behavior is independent of the h(0) in up to three layers thick films. At the same time, the initial surface roughness was shown to be essential for the flow-field-induced sequenced ordering of the microdomains. We identify a number of specific features of complex confinement on corrugated substrates as compared to flat space, such as extreme deviations from the bulk cylinder morphology to nonbulk lamella structures in thick films, as well as the violation of the surface tension-driven segregation of the soft PB component to the free surface. Differences in the flow behavior of PS-b-PB films and of PS homopolymer films under strong confinement, reported earlier, confirm the decoupling of the dynamics of the chains at the free surface from the underlying layers and suggest novel insights into the distribution of the surface fields through the film.