The aether was hypothesized to evenly permeate all space, providing a reference frame and medium for EM fields to travel through. EM fields require no such medium (to our knowledge).
There are many EM fields, and its hard to say if they are "moving" or at rest, you can say that some are constant (unchanging) and others are dynamic (changing) - which could be interpreted as they are moving.
There is no, one-EM field that permeates all space; instead there are a myriad separate fields that can overlap, interact, etc. (although they technically each extend to infinity unless blocked).