1) This question cuts to the core of what electromagnetism is. At any
nonzero velocity, you can't discuss magnetic fields separately from
electric fields.
2) As various people have pointed out, it's not clear what it means to
speak of a field that moves. If you rotate the *source* of a magnetic
field, such as a piece of magnetized iron, all sorts of weird effects
occur, such as centrifugal effects on the electrons in the iron,
centrifugal effects on the iron ions, plus the intrinsically
electromagnetic effects discussed below.
3) If I understand the spirit of the question correctly, the centrifugal
effects in the previous paragraph are unnecessary complications. If we
disregard centrifugal effects, then at any single point, circular motion is
indistinguishable from straight-line motion along the tangent at that
point. So let's discuss the straight-line case.
Accordingly, we can rephrase the question: imagine that in the lab from an
electron is moving through a magnetic field. At any given point, it feels
a force transverse to its motion. So the question is, what happens in some
other reference frame, such as one comoving with the electron?
Answer: The electron sees an electric field. You can easily calculate the
details of this field by Lorentz transforming the electromagnetic field
tensor from the one frame to the other.