Here's why you are confused Exnihiloest.
I agree that you can associate a frame to a space containing a field, but contrarily to what EM suggests, you can't say that the field has a speed v relative to this frame, nor is at rest in this frame.
Frames of reference are not associated to a SPACE, where did you learn that crap? Like I suggested get educated dude. Secondly, fields have a velocity.
If an electron is moving and it's field is not keeping up with it, what does that tell you? In ANY FRAME OF REFERENCE, (except accelerating ones, but we won't get into that yet since you haven't mastered the basics yet) the laws of electrodynamics hold just fine. Two electrons repel each other the same and Coulumbs law holds just fine. If the E fields did not keep up with these electrons, you wouldn't have this hold true.
It is a big misunderstanding to imagine a field with a speed i.e. to imagine a frame of reference in which the field would be "at rest".
You can't even agree with what your saying, your two statements above are contradictory. If it's a "big misunderstanding" to imagine a moving field, than it should not be a big misunderstanding to "
imagine a frame of reference in which the field would be at rest." I think there's some dissonance in your head due to your limited knowledge of electrodynamics.
So according to your imaginary laws, if fields can't move, than there must be an ABSOLUTE frame of reference, correct? I have news for you, This has been disproved and that's what relativity is all about.
This misunderstanding leads to a false paradox in Faraday motor (V in F=q.VxB is relative to the observer, not to the field, so no paradox).
Paradoxes exist for those that don't understand, however they are excellent opportunities to learn something new and more refined and improve existing theory.
When an electron is at rest, in respect to its own reference frame in which it is at rest, none magnetic field can move it because V=0 so F=q.VxB=0, but only an electric field. For induction, this field is E=-dA/dt, therefore a time variation is needed and from the viewpoint of the electron, the force it feels is F=q.E.
Like I said, F=qE is only half the equation. But let's work with your example here. FYI, I can move the electrons by moving a magnetic field across them, and it doesn't have to be non-uniform, it can be a UNIFORM magnetic field. You see it's all relative. the electron can be moving or the magnetic field. It's the relative motion that matters!
When an electron is not at rest (i.e. is moving at speed V relative to the frame of the observer, not of the magnetic field), then the observer see a force F=q.VxB acting onto the electron. But the electron doesn't see the force F=q.VxB: from its own reference frame,...
Do you think the electron cares WHO'S WATCH IT as a condition to DEVELOPING A FORCE OR NOT? Do you think an OBSERVER sees a FORCE on the electron, BUT THE ELECTRON DOES NOT? What kind of crap is that? Dude, if it moves relative to the magnetic field it will experience a force. Period! If not no force. So in your example, the electron moves but not relative to the magnetic field, so the magnetic field must be moving with it, so there is no force on it, no force EXPERIENCED BY THE ELECTRON AND NO FORCE PERCEIVED BY ANY OBSERVER. And by the way, forces are developed at 90 degrees to both the velocity and magnetic field, and an observer, moving or not, will see the side motion of the electron due to the force.
Exnihiloest, you are confused because you don't allow fields to move! Shame on you.
EM
PS. By the way, when i say moving I mean TRANSLATION. In the Homopolar generator, the magnetic field is not ROTATING about it's axis of symmetry. This is what confuses lots of people. In this case spinning the magnet or coil generating the mag field, about the symmetry axis accomplishes nothing perceived.