We calculate the flux diffusion into flat superconducting cables in AC magnetic fields applied on some part of the cable. At low frequencies, the transverse losses inside the field region are enhanced due to currents induced outside this region, whereas the losses are negligible in field-free regions. At high frequencies, the two contributions are approximately the same, being of the same type as in normal conductors (skin effect). The time constant for decay of the induced current is proportional to the cable length and can be much larger than calculated for spatially homogeneous AC fields. This could explain the existence of long-living oscillatory currents (or 'super coupling currents') in some cables for accelerator magnets.