It helps when starting out in relativity to recognize some rules which simplify 'new ways of thinking' relative to our everyday experiences. A guide which I found helpful was to remember when considering relativistic situations is that in the flat spacetime [no gravity] of special relativity the speed of light is always observed at 'c'...both locally and distant. Time and space may vary, but light is constant at speed 'c'.
But in general relativty, the curved characteristic of spacetime [gravity] can cause measures of distant phenomena to appear as if light moves at other than 'c'...for example it may move along a curved trajectory. But locally, where curvature can be ignored, light is always the same old 'c'. In other words, even if an observer measures some distant phenomena as appearing to have a lightspeed other than c, right there at the location of the phenomena, a local observer will record good old 'c'.
Would one expect the permativitity of free space to vary over the universe?
One would expect it to not vary...on the other hand, we have been fooled by Mother Nature on other occasions!
What we observe, each day, as older light than yesterday reaches us from the most distant parts of the universe, it is redshifted according to the same pattern as the prior day's light. If that pattern suddenly changed, we'd be tipped off there is something 'different' way out there where light originated some 13.8 billions years ago. So far it appears that as the distant particle horizon expands away from us, local characteristics there are the same as they are closer to us...redshift retains it's expected character.
all of these ideas are consistent with earlier posts here.