Thermal transpiration pumping in multistage assemblies is computationally investigated. Each stage is formed by combining in series-long microchannels with (a) uniform–uniform (“uni–uni”), (b) converging–uniform (“con–uni”), (c) diverging–uniform (“div–uni”) and (d) converging–diverging (“con–div”) rectangular cross sections. In all four investigated assemblies the generated pressure difference with the associated mass flow rate is fully assessed, in terms of inlet pressure, inclination parameter and number of stages. The analysis is based on linear kinetic modeling and is valid in the whole range of gas rarefaction. It is concluded that the “con–uni” and “div–uni” assemblies provide higher pressure differences and lower mass flow rates than the “uni–uni” assembly and they may be more suitable for specific pumping applications. The characteristics of the “con–uni” and “div–uni” assemblies are very close to each other. The multistage “uni–uni”, “con–uni” and “div–uni” assemblies are more stable when operating at small inlet pressures, where the pressure difference remains almost constant in wide ranges of mass flow rate. It is advisable to add as many stages as possible to increase, depending on the application, either the pressure difference or mass flow rate or both. Furthermore, the “con–div” assembly provides smaller pressure differences and mass flow rates than the other three, but it is suitable for diode applications. It is characterized by the so-called blocking inlet pressure, where the deduced pressure difference in the converging and diverging channels is the same. A detailed parametrization of the blocking inlet pressure in terms of inclination ratio, mean height, temperature difference and gas species, has been performed. It is concluded that multistage “con–div” assemblies may be ideally applied as thermally driven microfluidic diodes to control or block the flow, as well as to separate the species in multicomponent gas mixtures.