I'm wondering if a material's magnetic susceptibility or magnetic permeability would have an effect on the speed a magnetic field would propagate through it. I know it won't speed it up but I'm wondering if it could slow it down.
Think of a very long iron cylinder with an electromagnet on one end. It is quickly turned on and off once to send a magnetic pulse. Would this magnetic pulse travel slower than the speed of light?
I think it'd be similar to how light slows down in different materials.
Yes the magnetic properties will have an effect on the speed of the spread, and the phenomenon IS part of the description of how light slows down in materials. However, to see this phenomenon in this way means that we must use electromagnetic radiation of much lower frequency than visible light, because generally magnetic dipoles, owing to their relative size, react to electromagnetic fields much more slowly than electrons, hence almost always the effect is negligible at light frequencies. We also have the problem that most significantly magnetic materials are highly absorbing at visible light frequencies.
If we define the refractive index $n$ of a material to be the ratio of the freespace speed to in-material speed of light, if the material is linear and if the electromagnetic radiation is of the approriate frequency (as discussed above) then it follows directly from Maxwell's equations that:
$$n = \sqrt{\epsilon_r\,\mu_r}$$
where $\epsilon_r\,\mu_r$ are the material's relative electric and magnetic constant, respectively.
Magnetic effects are often highly nonlinear, which means the phenomenon is almost always much more complicated than discussed above. But the idea is the same: magnetic effects propagate through materials at lower than the vacuum lightspeed owing to the sloth of dipoles' reaction to the magnetic field.
If the material is working in the linear regime and loss is not significant, the effective refractive index is as I've given it above. However, with most magnetic materials, the conductivity is significant and you would need to use the general expression for the propagation constant from Maxwell's equations which is:
$$\gamma=\sqrt{i\,\omega\,\mu\,(\sigma + i\,\omega\,\epsilon)}$$
then the group velocity is $\mathrm{Re}(-i\,\frac{\mathrm{d}\gamma}{\mathrm{d}\omega})$.
