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The evolution of single stars at low metallicity has attracted a large interest, while the effect of metallicity on binary evolution remains still relatively unexplored. We study the effect of metallicity on the number of binary systems that undergo different cases of mass transfer. We find that binaries at low metallicity are more likely to start transferring mass after the onset of central helium burning, often referred to as case C mass transfer. In other words, the donor star in a metal poor binary is more likely to have formed a massive CO core before the onset of mass transfer. At solar metallicity the range of initial binary separations that result in case C evolution is very small for massive stars, because they do not expand much after the ignition of helium and because mass loss from the system by stellar winds causes the orbit to widen, preventing the primary star to fill its Roche lobe. This effect is likely to have important consequences for the metallicity dependence of the formation rate of various objects through binary evolution channels, such as long GRBs, double neutron stars and double white dwarfs.