Cosmic Background Radiation Question

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)?

Thanks,
Paul