  Metallicity is known to significantly affect the radial expansion of a
massive star: the lower the metallicity, the more compact the star, especially
during its post-MS evolution. We study this effect in the context of binary
evolution. Using the stellar-evolution code MESA, we computed evolutionary
tracks of stars at different metallicities, exploring variations of factors
known to affect the radial expansion (e.g. semiconvection, overshooting,
rotation). We find observational support for an evolution in which already at
metallicity $0.2Z_{\odot}$ massive stars remain relatively compact during the
Hertzprung-Gap (HG) phase and most of their expansion occurs during core-helium
burning (CHeB). Consequently, we show that metallicity has a strong influence
on the type of mass transfer evolution in binary systems. At solar metallicity,
a case-B mass transfer is initiated shortly after the end of MS, and a giant
donor is almost always a rapidly expanding HG star. At lower metallicity, the
parameter space for mass transfer from a more evolved CHeB star increases
dramatically. This means that envelope stripping and formation of helium stars
in low-metallicity environments occurs later in the evolution of the donor,
implying a much shorter duration of the Wolf-Rayet phase (even by an order of
magnitude) and higher final core masses. This metallicity effect is independent
of the impact of metallicity-dependent stellar winds. At very low
metallicities, a significant fraction of massive stars in binaries engages in
the first episode of mass transfer very late into their evolution, when they
already have a well-developed CO core. The remaining lifetime ($< 10^4$ yr) is
unlikely to be enough to strip the entire H-rich envelope. We also briefly
discuss the extremely small parameter space for mass transfer from massive
convective-envelope donors in the context of binary black hole merger
formation.
