  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 $1 M_{\odot}$ and $8 M_{\odot}$ 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 $q_{\rm qad}
\sim 0.25$ for stars crossing the Hertzsprung gap, while for convective giants
$q_{\rm qad}$ decreases from $\sim 1$ at the base of the RGB to $\sim 0.1$ at
the 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
$q > q_{\rm qad}$ 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.
