Stable mass transfer from a massive post-main sequence (post-MS) donor is thought to be a short-lived event of thermal-timescale mass transfer (∼ 10 −3 M yr −1 ) which within 10 4 yr strips the donor star of nearly its entire H-rich envelope, producing a hot, compact helium star. This long-standing picture is based on stellar models with rapidly-expanding Hertzprung gap (HG) donor stars. Motivated by a finding that in low-metallicity binaries, post-MS mass transfer may instead be initiated by donors already at the corehelium burning (CHeB) stage, we use the MESA stellar-evolution code to compute grids of detailed massive binary models at three metallicities: those of the Sun, the Large Magellanic Cloud (LMC, Z Fe;LMC /Z Fe; ≈ 0.36), and the Small Magellanic Cloud (SMC, Z Fe;SMC /Z Fe; ≈ 0.2). Our grids span a wide range in orbital periods (∼ 3 to 5000 days) and initial primary masses (10 M to 36-53 M , depending on metallicity). We find that metallicity strongly influences the course and outcome of mass-transfer evolution. We identify two novel types of post-MS mass transfer: (a) mass exchange on the long nuclear timescale (∆T MT 10 5 yr, Ṁ ∼ 10 −5 M yr −1 ) that continues until the end of the CHeB phase, and (b) rapid mass transfer leading to detached binaries with mass-losers that are only partially stripped of their envelopes. At LMC and SMC compositions, the majority of binary models with donor masses ≥ 17 M follow one of these two types of evolution. In neither (a) or (b) does the donor become a fully stripped helium star by the end of CHeB. Boundaries between the different types of post-MS mass transfer evolution are associated with the degree of rapid post-MS expansion of massive stars and, for a given metallicity, are sensitive to the assumptions about internal mixing. At low metallicity, due to partial envelope stripping, we predict fewer hot fully stripped stars formed through binary interactions as well as higher compactness of the pre-supernova core structures of mass losers. Nuclear-timescale post-MS mass transfer suggests a strong preference for metal-poor host galaxies of ultra-luminous X-ray sources with black-hole (BH) accretors and massive donors, some of which might be the immediate progenitors of binary BH mergers. It also implies a population of interacting binaries with blue and yellow supergiant donors. Partially-stripped stars could potentially explain the puzzling nitrogen-enriched slowly-rotating (super)giants in the LMC.