Why don't distant stars red/blue shift dramatically when we walk in different directions (due to the Rietdijk–Putnam argument)? https://en.wikipedia.org/wiki/Rietdijk%E2%80%93Putnam_argument

two people walking past each other in the street could have very different present moments. If one of the people were walking towards the Andromeda Galaxy, then events in this galaxy might be hours or even days advanced of the events on Andromeda for the person walking in the other direction

If this is true, then if I change direction while looking at the stars, shouldn't they redshift or blueshift abruptly by extreme amounts as the distant "now"?
If by changing my frame of reference, the thing I am observing has to "catch up" or "leave behind" several days worth of events, shouldn't this have something like a Doppler shift?
 A: 
If by changing my frame of reference, the thing I am observing has to "catch up" or "leave behind" several days worth of events, shouldn't this have something like a doppler shift?

For simplicity's sake we may say that an event in Andromeda is the creation of one EM-wave crest in Andromeda.
Those extra events/waves mentioned by the OP are needed, because more EM-wave crests are needed between the Andromeda and the observer, as the observer observes that all the waves between Andromeda and him become shorter.
Doppler-blueshift requires extra events/waves.
When events/waves at Andromeda are created according to the observer at a rate million times the normal rate, then the wave crests at Andromeda move according to the observer towards the observer at a rate million times the normal speed of light.
Then also the Andromeda itself moves according to the observer towards the observer at a rate million times the normal rate, where "normal rate" could be for example 1 m/s, if the observer has reached that walking speed.
So in our example Andromeda gets closer to the observer at rate 1000000 m/s, and light leaving Andromeda gets closer to the observer at rate 1000000 c. This means that if the observer walks slowly, then the length contraction of the distance between the observer and the Andromeda is very small compared to the length contraction of a light pulse connecting the Andromeda and the observer.
I guess that previous paragraph is incomprehensible. Well I'm trying to say that we can ignore the motion of the Andromeda according to the observer, unless the observer is very fast. And I'm also saying that Doppler blueshift is a length contraction of a light pulse. And one more thing that I'm saying is that one end of a thing moving faster than the other end is the same thing as length contraction.
Light beam contracts a lot, distance contracts by a small amount, a gap with no light would be formed, if the gap was not filled by more light  that must be emitted at very fast rate.
A: The only possible way one could get blueshift or redshift is because they are moving towards the galaxy rather than away from it (or vice versa). This blueshift is a very small effect, and compared to the redshift caused by the entire Earth moving away from Andromeda$^+$, this slight flick of blueshift from your frame does not matter, and cannot be measurable too.
Yes, what you see in Andromeda would actually differ based on whether you move towards or away from it. But, what would actually happens is, as you move towards Andromeda, you would observe the time there to be moving faster than what you would observe if you were moving away.
So, for example you may observe an event happening in 2 days, while the same event takes longer to happen for someone who is moving opposite (away from the galaxy).
This is what is meant by the 'advancing days' part. You would not immediately see days advancing, rather because time moves relatively faster to you, you would see the advancing days coming faster and earlier as compared to the 'away' person.$^*$ But, again, this is a very slight change, and compared to the movement of the Earth, this slight change in the flow of time is negligible.

$^+$ The Milky Way actually is headed towards Andromeda on a collision course. Anyway, the point of this statement was to say that the effect of blueshift caused by movement of the Earth is large enough, and the particular case of changing shifts is utterly negligible.
$^*$ Well, when you change direction, you would actually observe a 'jump' from one day to the next. See the Twins Paradox for more.
