Quote by Sherlock
The results of such comparisons are fairly clear. The clock whose state of motion has changed wrt the reference state accumulates time at a different rate. So, the average period of it's oscillator is different during the interval in which its average velocity wrt the reference state is different.
Ok, but we specify a reference state, so naturally we need to reassemble that state in order to make any meaningful comparisons between the clocks.
EDIT: I changed my mind about this. Because we specify a reference state involving a frame of reference in addition to the two clocks, then we don't need to bring the clocks back together in order to make meaningful statements about which clock is moving slower.
We of course do need to keep one clock in the reference state, and keep them both running, and continue tracking them both during the separation interval.

*IF* you make the additional assumption to use a specific reference frame's defintion of simultaneity, your logic is sound. But this additional assumption is required, and leads to a seeming paradox.
The seeming paradox is commonly called the "twin paradox", where A concludes that B's clocks are running slow, and B concludes that A's clocks are also running slow.
The key to resolving this paradox is to note specifically that one *does* have to make the extra assumption about what defintion of simultaneity one choses to use. There are as many different defintions of "simultaneous" as there are reference frames (which you call reference states). A's defintion of simultaneity is not the same as B's defintion. This is equivalent to my remark that the results of any clock comparision will depend on the details of the path by which the clock is transported. Until a reference frame is chosen (or a clock transport method is specified), the results of a direct comparison between the spatially separated clocks is not defined.
The recognition of the fact that there *are* different defintions of "simultaneous" is what prevents the twin paradox from actually being a paradox, i.e. it is necessary to recognize the relativity of simultaneity to understand how special relativity is selfconsistent.