The source and transport system of the Karlsruhe tritium neutrino experiment (KATRIN) must provide a significant reduction in tritium flow and gas density. It comprises a 10 m long windowless source tube, where the tritium gas is injected, followed by a differential pumping system and a cryogenic pumping system. The primary challenge of the analysis is that the gas flow changes from a viscous flow regime inside the source tube to a transitional flow regime at the first pumping stage, and to a molecular flow regime at the remaining stages of the differential pumping system and further downstream. A strong molecular beaming effect must be considered. This article presents the results of calculations of gas density and flow for the complete source and differential pumping system. It is shown that a total flow-rate reduction factor of 1.4x10(-8) can be attained, which is one of the prerequisites to achieve extreme-high vacuum conditions in the spectrometers used in the downstream end of the experiment.