The distinguishing feature of the evolution of close binary stars is the role played by the mass exchange between the component stars. Whether or not the mass transfer is dynamically stable is one of the essential questions in binary evolution. In the limit of extremely rapid mass transfer, the response of a donor star in an interacting binary becomes asymptotically one of adiabatic expansion. We use the adiabatic mass-loss model to systematically survey the thresholds for dynamical timescale mass transfer over the entire span of possible donor star evolutionary states. We also simulate mass-loss process with isentropic envelopes, the specific entropy of which is fixed to be that at the base of the convective envelope, to artificially mimic the effect of such mass loss in superadiabatic surface convection regions, where the adiabatic approximation fails. We illustrate the general adiabatic response of 3.2 M-circle dot donor stars at different evolutionary stages. We extend our study to a grid of donor stars with different masses (from 0.1 to 100 M-circle dot with Z = 0.02) and at different evolutionary stages. We proceed to present our criteria for dynamically unstable mass transfer in both tabular and graphical forms. For red giant branch (RGB) and asymptotic giant branch (AGB) donors in systems with such mass ratios, they may have convective envelopes deep enough to evolve into common envelopes on a thermal timescale, if the donor star overfills its outer Lagrangian radius. Our results show that the RGB and AGB stars tend to be more stable than previously believed, and this may be helpful to explain the abundance of observed post-AGB binary stars with an orbital period of around 1000 days.