Context. The origin and number of the Galactic supergiant X-ray binaries is currently not well understood. They consist of an evolved massive star and a neutron star or black-hole companion. X-rays are thought to be generated from the accretion of wind material donated by the supergiant, while mass transfer due to Roche-lobe overflow is mostly disregarded because the high mass ratios of these systems are thought to render this process unstable. Aims. We investigate how the proximity of supergiant donor stars to the Eddington limit, and their advanced evolutionary stage, may influence the evolution of massive and ultra-luminous X-ray binaries with supergiant donor stars (SGXBs and ULXs). Methods. We constructed models of massive stars with different internal hydrogen and helium gradients (H/He gradients) and different hydrogen-rich envelope masses, and exposed them to slow mass-loss to probe the response of the stellar radius. In addition, we computed the corresponding Roche-lobe overflow mass-transfer evolution with our detailed binary stellar evolution code, approximating the compact objects as point masses. Results. We find that a H/He gradient in the layers beneath the surface, as it is likely present in the well-studied donor stars of observed SGBXs, can enable mass transfer in SGXBs on a nuclear timescale with a black-hole or a neutron star accretor, even for mass ratios in excess of 20. In our binary evolution models, the donor stars rapidly decrease their thermal equilibrium radius and can therefore cope with the inevitably strong orbital contraction imposed by the high mass ratio. We find that the orbital period derivatives of our models agree well with empirical values. We argue that the SGXB phase may be preceded by a common-envelope evolution. The envelope inflation near the Eddington limit means that this mechanism more likely occurs at high metallicity. Conclusion. Our results open a new perspective for understanding that SGBXs are numerous in our Galaxy and are almost completely absent in the Small Magellanic Cloud. Our results may also offer a way to find more ULX systems, to detect mass transfer on nuclear timescales in ULX systems even with neutron star accretors, and shed new light on the origin of the strong B-field in these neutron stars.