That makes perfect sense, it's how induction works, when a conductor is moved through an H-field a potential differance is induced in it causing an E-field to develop and the reverse is true with a static electric field. If you're standing still then the steady state H or E field will be constant but if you move nearer or further away from it, it varies as far as you're concerned anyway.
You are confusing a time varying H field with a relativisticly transformed H field. Induction, A.K.A. Faraday's law, is Maxwell's third equation. The equations of relativity are what keeps all Maxwell's equations invariant under any velocity frame of reference.
This example should illustrate the difference.
Induction: You move a wire ring around a magnet. This causes the magnetic flux through the wire ring to change with time, in turn causing a "curled" E-Field in the wire ring, generating an electric current. The wire ring is moving, so
it is experiencing the field at differerent distances from the magnet as it moves causing a time varying H field from the perspective of the wire ring.
Relativity: The wire ring is held
stationary three inches away from the magnet. The magnetic flux does not change.
Billy is standing next to the magnet and "sees" only a constant, static magnetic field in the center of the ring.
Mary is moving at half the speed of light, flying by the magnet and ring. She looks at the center of the ring, and "sees" a constant, static, non-time varying magnetic
and electric field. This electric field is not curled (because relativistic transforms are linear) and does not induce a current in the ring. Mary is not looking at the magnetic field at her position (which would definitely be changing because her position is changing, just as the ring's position is changing in the Induction example). We are talking about the static fields at a fixed distance away from the magnet.
Joe is moving at 3/4 the speed of light, and sees a different combination of static, non-time varying magnetic and electric fields at the center of the ring than Mary or Billy, because his frame of reference has a different velocity.
