The all-flow-regime model of fluid flow, previously applied in  to flows with axially and temporally uniform Reynolds numbers, has been implemented here for flows in which the Reynolds number may either vary with time or along the length of a pipe. In the former situation, the timewise variations were driven by a harmonically oscillating inlet flow. These oscillations created a succession of flow-regime transitions encompassing purely laminar and purely turbulent flows as well as laminarizing and turbulentizing flows where intermittency prevailed. The period of the oscillations was increased parametrically until the quasi-steady regime was attained. The predicted quasi-steady friction factors were found to be in excellent agreement with those from a simple model under which the flow is assumed to pass through a sequence of instantaneous steady states. In the second category of non-constant-Reynolds- number flows, axial variations of a steady flow were created by means of a finite-length conical enlargement which connected a pair of pipes of constant but different diameters. The presence of the cross-sectional enlargement gives rise to a reduction of the Reynolds number that is proportional to the ratio of the diameters of the upstream and the downstream pipes. Depending on the magnitude of the upstream inlet Reynolds number, the downstream fully developed flow could variously be laminar, intermittent, or turbulent. The presence or absence of flow separation in the conical enlargement had a direct effect on the laminarization process. For both categories of non-constant-Reynolds-number flows, laminarization and turbulentization were quantified by the ratio of the rate of turbulence production to the rate of turbulence destruction.