If you open a switching step down converter like the one used in your desktop computer, you will only find that a relatively small torroidal transformer ultimately provide the power to the user.
Sure, the flyback operates at high frequency, which raises the impedance enormously, and there is a whole regulation mechanism, but that's not the point: These PSU can easily provide 7A or much more (15A not unusual). This high current is ultimately provided by the small transformer. Without the magnetic effect, nothing would happen.
My question is: how these transformers can handle such high a current without saturating?