I don't get your problem here - as it says in the bit you quoted from Wiki: "...Gravitational redshift and blueshift frequency ratios are the inverse of each other, suggesting that the "seen" frequency-change corresponds to the actual difference in underlying clockrate."
It's just a matter of relative rates. The observer further out inteprets the clock further in as time dilated (running slow = redshifted). Conversely, an observer further in interprets the clock further out as time contracted (running fast = blue-shifted). Substitute light source = oscillator for 'clock', and there it all is.
Sorry, the problem is the use of the word "Doppler."
There are three types of true "Doppler" effects, Universe expansion source-observer velocity differentials, ordinary source-observer velocity differentials, and bending of EM by gravitational space-time distortions.
There is indeed a gravitational redshift, which is seen in things like white dwarfs.
It is certainly due to large distortions in space-time of some sort.
That is NOT a Doppler shift. That's my point.
But there is indeed a Doppler shift that can be caused by gravity, too.
That famous radio wave experiment showed that pulses take longer and longer to reflect back when the beam travels closer and closer to the sun.
Since the speed of light is constant in a vacuum, no matter what, that can only mean one thing - the path is longer, i.e. an apparent velocity differential.
That time delay causes a Doppler shift. The delay is longer than would be predicted by GR geometry alone, but that's irrelevant, it's still a Doppler shift caused by geometry. It is exceedingly tiny but not non-existent. That's because it appears as if there is relative movement, even though there isn't - the source and observer don't intrinsically "know" that the beam has passed through a portion of distorted space-time.
In fact, Einstein's first prediction of how much the sun would bend star light was off by a factor of two. His geometry was off. His geometry changed until if fit the facts. He threw in the "cosmological constant" for the same reason, by his own admission.
Gravitational lensing also causes a Doppler shift.
It's exceedingly tiny as well, but it's there.
Neither one of these two examples of gravitational Doppler shifts can possibly be blue-shifting because neither case can result in a shorter apparent path.
Einstein's thought experiments were interesting, but there haven't been too many more since GR that have resulted in advanced theories.
In fact, Einstein spent a lot of the rest of his life trying to think up experiments that disproved QM.
What does any of this have to do with "slices of now"?
A lot, IMO. What is the minimum possible Doppler shift?
Plank time? If so, that's your "slice of now."
But what if you add time dilation, then what?
Yes, I am being an ash. LOL
Heck, I could have brought up the quasar red-shift problem to become a double ash.