The startup flow of thixotropic yield stress fluids in tubes is studied numerically. In the situation of interest, the structured fluid is displaced by another fluid, under the application of a constant inlet pressure. The structured fluid is assumed to obey a recently proposed model for thixotropic yield-stress fluids that relies on a structure parameter. The momentum conservation principle is employed in a simplified form that arises by assuming quasi-steady flow, i.e. the shear stress radial distribution is assumed to be linear as it is in a steady flow. Inertial effects are also included in the analysis. A numerical procedure was developed to solve the differential equations that arise in the problem. The time evolutions of the average velocity and of the radial distribution of the structure parameter are obtained for different combinations of the rheological and flow parameters. Among other features, the approach is capable of predicting the avalanche effect, i.e. situations of no flow for long time periods and then the sudden onset of motion.