Is it physically possible to substantially change a large magnetic field with a smaller one analogous to the electric field at the gate being able to partially or completely block the flow from source to drain in a transistor? I'm imagining being able to reorient the field from a rare earth magnet using less powerful electromagnets - would this currently be just science fiction?
Almost. There is a device called a magnetic amplifier in which a small magnetic field is used to control a much larger field by means of a cleverly-designed transformer. At one time it was thought that these could be used in electric power control systems but cheap and reliable transistors took over the market before that happened.