The stability of mass transfer is critical in determining pathways towards various kinds of compact binaries, such as compact main-sequence white-dwarf binaries, and transients, such as double white-dwarf mergers and luminous red novae. Despite its importance, only very few systematic studies of the stability of mass transfer exist. Using the 1D stellar evolution code MESA, we study the behaviour of mass-losing post-main-sequence donor stars with masses between 1M_{sun} and 8M{sun} in binaries, without assuming that the donor star responds to mass loss adiabatically. We treat the accretor as a point mass, which we do not evolve, and assume the mass transfer is conservative. We find that the criterion that best predicts the onset of runaway mass transfer is based on the transition to an effectively adiabatic donor response to mass loss. We find that the critical mass ratio q_qad~0.25 for stars crossing the Hertzsprung gap, while for convective giants q_qad_ decreases from ~1 at the base of the RGB to ~0.1 at the onset of thermal pulses on the AGB. An effectively adiabatic response of the donor star only occurs at a very high critical mass-transfer rate due to the short local thermal timescale in the outermost layers of a red giant. For q>q_qad_ mass transfer is self-regulated, but for evolved giants the resulting mass-transfer rates can be so high that the evolution becomes dynamical and/or the donor can overflow its outer lobe. Our results indicate that mass transfer is stable for a wider range of binary parameter space than typically assumed in rapid binary population synthesis and found in recent similar studies. Moreover, we find a systematic dependence of the critical mass ratio on the donor star mass and radius which may have significant consequences for predictions of post-mass-transfer populations.