Inertial reference frames are not time-dependent. I would say that "all inertial reference frames" include all such frames in all times. The recent debate about the varying speed of light stems from the fine structure factor α and cosmological problem:
α =
e2
ħ c
where ħ = h/(2π), h: Planck's constant, e: charge of an electron, c: speed of light.
As long as the fine structure constant is concerned, we can't tell whether the speed of light is varying because of two different results so far. I need to have a look at a paper (Physical Review D, vol 69, p121701). As for cosmological stuff, I have no idea unless I go through the details (whether the speed of light is a constant but it changes in a certain condition).
