A very specific question I have an answer to. No it doesn’t.
Using PyEphem in PyLunar (because I am a rebel like that), let’s look at right ascension values through the course of three months for two timezones and a variety of locations.
There is a series of data you want to look at to be sure that you don’t have to make extra effort to solve this.
How I do things
My code is pretty much like this:
import datetime, pylunar
for single_date in (datetime.datetime.today().date() + datetime.timedelta(n) for n in range(365)):
_today = datetime.datetime.combine(single_date, datetime.time(16, 0))
mi = pylunar.MoonInfo(location)
mi.update(single_date)
ra = degree_to_hms(mi.ra())
fra = float(ra.split()[0] + '.' + ra.split()[1] + ra.split()[2].split('.')[0])
print('[%s] %s' % (single_date, fra))
The only function I will leave outside of this is degree_to_hms – it is an algorithm rather easy to find, I have nothing to do with it. You can find it easily.
Within one timezone
So what is going on – I am iterating a year from today and get right ascension. Time is 4pm (16:00) every day. And location
is now set to (55, 75), (37, 61) – it is Moscow, my hometown.

It is apparent what is going on – every day right ascension is acquiring 1 hour and +- several minutes.
If we check how do the right ascension change for the same longitude but different latitude – it is a lazy way to emulate how does right ascension vary in different places in the same timezone. Looking at the previous picture, it’s not too hard to guess how things will enfold.

Apprently, most days right ascension will be pretty much the same. But on some days there will be a drastic difference. Pulling out graphics, you will see that at some hours of some days (end of moon cycle, per se) for some locations the Moon has already started the new cycle, and for some it didn’t. So some already have 0 hours something minutes right ascension, and some are still at 23 hours.

So within the timezone things are close, but across the world?
Further experiments lead me to believe that at the same hour, but in different locations at the same time the situation is pretty much the same. For London and for Tokyo at 3pm you will get approximately same numbers.

Again, they vary, but not by much.