Wasn't it Paul Jones who wrote:
Hi,

The following question has been troubling my tiny brain, please could
one of you boffins answer it for me:

If the Cosmic Background Radiation is leftover from the Big Bang,
which is also the source of all the galaxies then, given that almost
all the galaxies are moving away from us, why isn't the CBR also
moving away from us (in which case, according to my underdeveloped
noodle, should mean that we can't detect it)?

Relativity.

Electromagnetic radiation always travels at the speed of light relative
to any observer. So radiation that set off from the big bang 13.7
billion years ago arrives at the speed of light. In exactly the same way
that light from a star arrives at the speed of light, even if the star
is travelling away from us at half the speed of light.

The speed of the em-radiation is constant, but the speed difference
between source and observer causes the wavelength to be shifted. In the
case of the cosmic background, the red shift is so great that it's gone
right though the visible wavelengths and down into the microwave region.

--
Mike Williams
Gentleman of Leisure