Here is a basic look at a field coil driver:
https://www.edn.com/field-coil-speakers-obsolete-or-the-future/
The field coil is separate from the voice coil and does not move. As
@Philbo King mentioned above, the field coil also served as the DC choke (power supply filter) for tube amplifiers, which meant they had to be insulated to handle several hundred (at least) volts. I, and probably many others, have seen the aftermath of the field coil insulation breaking down and turning the driver in to a charred mess, including catching the (paper) cone itself literally on fire. I would expect, but do not know, that today's field coils would operate at much lower voltage and higher current, and probably not feed the amplifier (though see below).
Multiple voice-coil windings have been used for many years; the first time I ran into them was on the Infinity IRS system's woofers that had two windings to help control the response. I used one in my first DIY subwoofer but used the second coil as a feedback source for a servo system. A lot of woofers these days have two coils so the speaker designer (that would
not be me) can trade impedance and sensitivity, etc. by how the two windings are used (connected and driven).
A field coil design could offer a lot of options for parameter control, but also adds a number of additional variables that must all work together, at a significant cost in complexity and power. I have a vague memory of some papers discussing dynamically varying the magnetic field with the signal to optimize power and distortion, basically turning the DC power supply into an active audio amplifier driving the field coil, but do not know if anything made it to the commercial product phase. Another vague memory is of a paper showing field coils on either side of the voice coil to create a sort of push-pull field; again, no idea if a practical commercial example was ever created. You could use the field coil to power the amplifier, as done in the past (for tubes), thus varying the field coil current (and thus magnetic field) with amplifier output level to provide an interesting feedback loop to reduce standing power. All sorts of interesting possibilities, but whether really worthwhile, I could not say (but that they are barely a niche now says something in my mind).
Maybe I'll go fire up my old spark-gap transmitter to create a signal for my crystal radio now. The arcs from the transmitter create a very wideband signal so must be better than the current band-limited AM stations, don't you think?
Edit: I'll defer to
@fpitas -- I was typing and held in place for a while and he answered (fielded?
) the questions more thoroughly than I.