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I presume that the photons from the CMB approach the earth from all directions, otherwise we couldn't detect them with a picture where it is present everywhere in the universe with a tiny anisotropy.

Now the CMB is a result of the big bang. Another result of the big bang is that in all directions we see galaxies moving away from us, like we are in the center of the universe (which is probably not the case).

But why are all the galaxies (except a few) moving away from us while the CMB is approching us. What is the reason for that difference? Or aren't those photons coming from everywhere?

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The Big Bang happened everywhere and the recombination of electrons with nuclei - the thing that causes the CMB - also happened everywhere. So in every direction you look you can see the radiation from that recombination.

But that recombination also happened a long time ago and so, given light has a constant velocity, the places where we can see the recombination take place are also a long way away. And, as a result of the expansion of space-time, that radiation is very red-shifted, so instead of the recombination looking like it is happening at roughly 4000 degrees Kelvin, it looks like something happening at 3 degrees Kelvin.

That expansion is also what makes it look like the distant galaxies are all moving away from us and the further away the faster they seem to be moving.

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You are comparing apples and oranges. The CMB is like the light from a distant galaxy, which while the galaxy is receding the light approaches us. CMB is the light emitted by universe at recombination. The matter that emitted the CMB is/was receding from us with a very high red-shift.

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