Laboratory experiments with rock samples show that transient creep, at which strain grows with time and strain rate decrease at constant stress, occurs while creep strains are sufficiently small. The transient creep at high temperatures is described by the Andrade rheological model. Since plate tectonics allows only small deformations in lithospheric plates, creep of the lithosphere plates is transient whereas steady-state creep, described by non-Newtonian power-law rheological model, takes place in the underlying mantle. At the transient creep, the effective viscosity, found in the study of postglacial flows, differs significantly from the effective viscosity, which characterizes convective flow, since timescales of these flows are very different. Besides, the transient creep changes the elastic crust thickness estimated within the power-law rheology of the lithosphere. Two problems of convective stability for the lithosphere with the Andrade rheology are solved. The solution of the first problem shows that the state, in which large-scale convective flow in the mantle occurs under lithospheric plates, is unstable and must bifurcate into another more stable state at which the lithospheric plates become mobile and plunge into the mantle at subduction zones. If the lithosphere had the power-law fluid rheology, the effective viscosity of the stagnant lithospheric plates would be extremely high and the state, in which large-scale convection occurs under the stagnant plates, would be stable that contradicts plate tectonics. The mantle convection forms mobile lithospheric plates if the effective viscosity of the plate is not too much higher than the effective viscosity of the underlying mantle. The Andrade rheology lowers the plate effective viscosity corresponding to the power-law fluid rheology and, thus, leads to instability of the state in which the plates are stagnant. The solution of the second stability problem shows that the state, in which the lithospheric plate moves as a whole with constant velocity, is stable but small-amplitude oscillations are imposed on this motion in regions of thickened lithosphere beneath continental cratons (subcratonic roots) where the thickness of the lithosphere is about 200 km. These oscillations create small-scale convective cells (the horizontal dimensions of the cells are of the order of the subcratonic lithosphere thickness). Direction of motion within the cells periodically changes (the period of oscillations is of the order of 108 yr). The small-amplitude convective oscillations cause small strains and do not destroy the thickening of the lithosphere beneath cratons. Thus, the transient creep of the lithosphere explains not only mobility of the lithospheric plates but longevity of subcratonic roots as well.