What would the CMB look like somewhere else in the universe? An answer to a question on Worldbuilding.SE opined that if you were suddenly transported an unknown distance and direction away by magic you could use the Cosmic Microwave Background to determine your position relative to the Earth.

This lead me to wonder, what does the CMB look like at different positions in the universe? Can changes in how the CMB looks be used to determine your position, direction, or velocity relative to Earth?
The CMB is nearly featureless, but the operative word is nearly. The tiny temperature fluctuations that we've measured should look about the same anywhere within tens to hundreds of millions of light years of Earth or so. This would allow you to get an absolute orientation relative to the CMB and thus relative to Earth.

It's fairly easy to measure velocity relative to the CMB, so if you measure your velocity relative to the CMB somewhere else (within that 10-100 million lightyears of Earth, anyway) and if you know your orientation, you can compute your velocity relative to Earth.

But I don't see any way to easily measure the distance from Earth except very approximately. (The CMB that you see from any point is different because it comes from a different sphere. If you move far enough from Earth, the temperature fluctuations you see are no longer the ones we see from Earth because you're sampling a different part of the 300-year-old universe.)

Later:
I've been thinking about the position problem and there's a solution, but it's complicated.  The CMB samples the universe of 13.7 billion years ago and from any point in it see the distribution of matter in a shell that old. If we move to another point, in effect, the shell shifts also and samples different parts of the gas. The key point is that the faint ripples in the gas's temperature that we see as mottling in the CMB are three-dimensional bubbles.

So if you shift a short distance in space, the CMB will shift the most in the direction of motion (both in front and behind) and the least at right angles. Furthermore, the amount of shift will depend on the angular size of the fluctuation you're looking at. Low-angular-frequency fluctuations come from physically large ripples int he gas and you need to move a long distance to move the shell you are sampling out of the bubble. High-angular-frequencies ripples come from smaller bubbles and shorter shifts in position will take the shell you observe out of one bubble and into another.

Waving my hands just a tiny bit, it should be possible to use statistical methods to estimate the amount and direction of movement by comparing two detailed CMB maps. A major limitation of the method is that the finer the angular resolution, the sooner you lose all correlation between the two views fo the CMB, but the coarser the angular resolution, the coarser your measurement will be.

Numbers? As long as we're not talking about jaunteing many billions of light years, we can ignore general relativistic issues. A fluctuation with an angular size of 1 degree corresponds to a CMB fluctuation which appears to us right now to be around 250 million light years across and consequently ought have a good chance to be visible for moves within roughly that distance. This suggests that the CMB method ought to be able to give you a rough position up to maybe a half-billion or even a billion light years, but will drop off rapidly in accuracy after that.

Note that for shorter hops, it ought to be possible to actually recognize galaxy super-clusters and get a position that way.