In many practical circumstances, the flow in microchannels is driven by moving surfaces that impart shear to the fluid. The shear may generate a pressure differential or the pressure may be imposed to affect the resulting flow and heat transfer. Processes where such flows arise include coating, cooling and thermal treatment of moving wires, fibers, and microscale devices. One particularly important circumstance is the optical fiber coating process, where both the entrance of the moving surface into a reservoir of fluid, as well as the exit, are of interest, since the thermal transport in the relevant microchannels influences the resulting coating very substantially. Similarly, the drawing of microscale fibers and the cooling of fibers after drawing are important in the overall manufacturing process and involve microchannel transport. This paper discusses the basic considerations in such processes, particularly the flow that arises in drawing and coating and the menisci that are observed at the inlet and outlet regions of the two microchannels. Experimental and analytical/numerical work has been carried out on these flows and the results obtained are presented. An important aspect is the pressure rise in the channel for narrowing flow domains, such as those employed in dies, and a comparison with imposed pressures. It is found that, in many practical problems, the shear generates much higher pressures than the typically imposed pressures and, thus, the flow is largely dominated by the shear effects due to the moving surfaces. It is also found that the flow develops very rapidly in forced convective cooling by gases, such as nitrogen and helium, resulting in thermal transport in a largely developed flow region. Comparisons between experimental and numerical results show fairly good agreement, indicating the validity of the model for such complex microchannel flows.