That's not a good way of describing the actual law in a curved spacetime, i.e., a spacetime that contains gravitating masses. In such a spacetime there is no single global definition of "speed"; you can't compare speeds at spatially separated points.

A better way to state the law is that the light cone structure of the spacetime constrains the motion of all bodies: timelike bodies move within the light cones, lightlike bodies (like light itself) move exactly on the light cones. But once you state it that way, it becomes obvious that this law does not impose any constraints on "time/space fluctuations".

The "twin paradox" [1] is a prime example. The two twins depart from a common point in time and space, go about their separate travels, and meet again at a common point in space-time. Despite both twins always having the same constant speed of light, one of the twins takes a shorter path through time to get to the meeting point--one twin aged less than the other. In the paradox case, the shorter/longer paths are due to differences in acceleration. But the same thing happens due to differences in gravitation along two paths. (In fact, IIUC, acceleration and gravitational differences are the same thing.)

Just thinking about the math makes my head hurt, but it's apparent that two different photons can have taken very different journeys to reach us. For example, the universe was much denser in the dim past. Old, highly red-shifted photons have spent a lot of time slogging through higher gravitational fields. As a layman, that would suggest to me that, on average, time would have.. moved slower for them?... they would be even older than naive appearances suggest. I don't think the actual experts are naive, so that's been accounted for, or there's confounding factors. But I could also imagine that more chaotic differences, such as supernovas in denser galatic centers vs. the suburbs, or from galaxies embedded in huge filaments, could be hard to calculate.

[1] https://en.wikipedia.org/wiki/Twin_paradox