Electron transmission through chiral molecules induced by circularly polarized light can be very different for mirror-image structures, a peculiar fact given that the electronic energy spectra of the systems are identical. We propose that this asymmetry--as large as 10% for resonant transport--arises from different dynamical responses of the mirrored structures to coherent excitation. This behavior is described in the context of a general novel phenomenon of current transfer (transfer of charge with its momentum information) and accounts for the observed asymmetry and its dependence on structure.