The temporal evolution of drift Alfvén waves in an inhomogeneous plasma of low and finite pressure with homogeneous shear flow is studied as an initial value problem without the use of spectral expansion in time. The cases of plasma with cold and hot ions, weak and strong flow shear are considered separately. It is shown that the conventional modal structure of the stable and unstable drift and Alfvén waves holds only for a limited time in the initial stage of its evolution. For larger times, nonmodal effects due to the velocity shear define the development of drift Alfvén waves and drift Alfvén instabilities. For the regimes of low flow shear, which corresponds to the period of the low-to-high transition, the long time evolution of these instabilities as well as their saturation are determined by the nonlinear effects such as the nonlinear decorrelation effect. In contrast, the plasma with strong flow shear, which corresponds to the regime of the developed transport barriers, is stable against the development hydrodynamic drift Alfvén and resistive drift Alfvén instabilities. The frequency increase caused by the shear flow brings the Alfvén wave phase speed close to the electron thermal speed where strong electron Landau damping occurs. At this stage, a kinetic approach for the description of these waves becomes necessary.