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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 which strips the donor star of nearly its entire H-rich envelope, producing a hot helium star. This long-standing picture is based on stellar models with Hertzprung gap (HG) donors. Motivated by a finding that in low-metallicity binaries, post-MS mass transfer may instead be initiated by core-helium burning (CHeB) donors, we use the MESA code to compute grids of detailed massive binary models at three metallicities: Solar and that of the Large and the Small Magellanic Cloud (LMC, SMC). We find that metallicity strongly influences the course and outcome of mass transfer. We identify two novel types of post-MS mass transfer: (a) mass exchange on the long nuclear timescale 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. 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 are associated with the degree of rapid post-MS expansion and, for a given metallicity, are sensitive to the assumptions about internal mixing. At low metallicity, we predict fewer hot fully stripped stars formed through binary interactions and higher compactness of pre-supernova cores. 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., Comment: Accepted for publication in A&A. Stellar tracks and MESA inlists available at zenodo.org/record/6412508

Binary neutron stars have been observed as millisecond pulsars, gravitational-wave sources, and as the progenitors of short gamma-ray bursts and kilonovae. Massive stellar binaries that evolve into merging double neutron stars are believed to experience a common-envelope episode. During this episode, the envelope of a giant star engulfs the whole binary. The energy transferred from the orbit to the envelope by drag forces or from other energy sources can eject the envelope from the binary system, leading to a stripped short-period binary. In this paper, we use one-dimensional single stellar evolution to explore the final stages of the common-envelope phase in progenitors of neutron star binaries. We consider an instantaneously stripped donor star as a proxy for the common-envelope phase and study the star's subsequent radial evolution. We determine a range of stripping boundaries which allow the star to avoid significant rapid re-expansion and which thus represent plausible boundaries for the termination of the common-envelope episode. We find that these boundaries lie above the maximum compression point, a commonly used location of the core/envelope boundary. We conclude that stars may retain fractions of a solar mass of hydrogen-rich material even after the common-envelope episode. If we consider orbital energy as the only energy source available, all of our models would overfill their Roche lobe after ejecting the envelope, whose binding energy includes gravitational, thermal, radiation, and recombination energy terms., 14 pages, 10 figures, 3 tables. Accepted for publication in MNRAS