electromagnetism - Moving magnetic fields?

I was reading through Feynman Lectures on physics vol II when I came across this paragraph which I don't quite seem to understand:

Since electric and magnetic fields appear in different mixtures if we change our frame of reference, we must be careful about how we look at the fields E and B. For instance, if we think of “lines” of E or B, we must not attach too much reality to them. The lines may disappear if we try to observe them from a different coordinate system. For example, in system S′ there are electric field lines, which we do not find “moving past us with velocity v in system S.” In system S there are no electric field lines at all! Therefore it makes no sense to say something like: When I move a magnet, it takes its field with it, so the lines of B are also moved. There is no way to make sense, in general, out of the idea of “the speed of a moving field line.” The fields are our way of describing what goes on at a point in space. In particular, E and B tell us about the forces that will act on a moving particle. The question “What is the force on a charge from a moving magnetic field?” doesn’t mean anything precise.

But later in another chapter on induced currents he talks about how moving a wire perpendicular to a magnetic field produces a force on the electrons which cause them to move along the wire, generating a current. But he also mentions that moving the magnet itself(and hence it's magnetic field) in the direction opposite to the direction the wire was previously moved produces the same force on the electrons.

Answer

You say "and hence the magnetic field" moves. No it doesn't, and that's exactly the point Feynman is trying to make.

Read it carefully. He considers the magnetic field to be a property of space, so to speak. At each point in space there is a magnetic field. It's magnitude and direction can change, but the field at that point is fixed to live at that point. The field at a point can change, and the field at the point's neighbors can change in such a way that the pattern of the field moves (a wave). But the field itself does not move.