  Close stellar binaries are prone to undergo a phase of stable mass transfer
in which a star loses mass to its companion. Assuming that the donor star loses
mass along the instantaneous interstellar axis, we derive the orbit-averaged
equations of motion describing the evolution of the donor rotational angular
momentum vector (spin) which accompanies the transfer of mass. We consider: (i)
a model in which the mass transfer rate is constant within each orbit and (ii)
a phase-dependent rate in which all mass per orbit is lost at periapsis. In
both cases, we find that the ejection of $\gtrsim 30$ per cent of the donor's
initial mass causes its spin to nearly flip onto the orbital plane of the
binary, independently of the initial spin-orbit alignment. Moreover, we show
that the spin flip due to mass transfer can easily dominate over tidal
synchronisation in any giant stars and main-sequence stars with masses
$\sim1.5$ to $5\,\rm M_\odot$. Finally, the general equations of motion,
including tides, are used to evolve a realistic population of massive binary
stars leading to the formation of binary black holes. Assuming that the stellar
core and envelope are fully coupled, the resulting tilt of the first-born black
hole reduces its spin projection onto the orbit normal by a factor
$\sim\mathcal{O}(0.1)$. This result supports previous studies in favour of an
insignificant contribution to the effective spin projection, $\chi_{\rm eff}$,
in binary black holes formed from the evolution of field binaries.
