cmb - Does the cosmic microwave background change over time?

The CMB patterns do indeed change over time, although statistically they remain the same, and although it will not be noticeable on human timescales.

The CMB we observe now comes from a thin shell with us in the center, and with a radius equal to the distance that the light has traveled from the Universe was 379,000 years old and until now. As time passes, we will receive CMB from a shell with an increasingly larger radius. As that light has traveled farther through space, it will, as you say, be more redshifted, or "cooler". But it will also have been emitted from more distant regions in the early Universe that, although statistically equivalent, simply will be other regions and hence look different.

The patterns that change the fastest are the smallest patterns we can observe. The angular resolution of the Planck satellite is 5-10 arcmin. Since the CMB comes from a redshift of ~1100, the angular diameter distance defining the physical distance spanned by a given angle — is ~13 Mpc, so 5 arcmin corresponds to a physical scale of roughly 19 kpc in physical coordinates, or 21 Mpc in comoving coordinates (that is, a structure spanning 5 arcmin today were ~19 kpc across at the time of emission, but have now expanded to a size of ~21 Mpc, with 1 Mpc = 1000 kpc = 3261 lightyears).

Assuming an isotropic Universe, if the smallest observable parcels of gas were 19 kpc across perpendicular to our line of sight, they are also on average 19 kpc across along our line of sight.

So the question of how fast the CMB changes comes down to how much time did it take light to travel 19 kpc when the Universe was 379,000 years old. This is not simply 19 kpc divided by the speed of light, since the Universe expands as the light travels, but it's pretty close. If my calculations are not wrong, we will have to wait until the CMB has been redshifted roughly to 1200, which will take around 60,000 years (assuming that Planck will not get replaced by better instruments within that time which, um, is dubious).

So you're right, you could make a 3D image of the CMB, but since the patterns are much larger than a lightyear, you don't have to take a new picture every year.

Actually, I'm a bit surprised by these small numbers of 19 kpc and 60 kyr. If somebody can spot a mistake here (e.g. a missing (1+z) factor), please correct. My calculations are here (in Python):

import numpy as np
import cosmolopy.distance as cd
import cosmolopy.constants as cc

cosmo   = {'omega_M_0':0.27, 'omega_lambda_0':0.73, 'omega_k_0':0.0, 'h':0.7}
zCMBnow = 1100.
dA      = cd.angular_diameter_distance(zCMBnow, **cosmo) # In Mpc
kpc_5am = dA*1e3 * np.pi/(180.*3600) * 5*60              # Scale at source in kpc per 5 arcmin
print '5 arcmin corresponds to ' + str(round(kpc_5am,1)) + ' kpc at the source'

zCMBfuture = 1200.
d          = cd.light_travel_distance(zCMBfuture, zCMBnow, **cosmo) * 1e3
print 'When the CMB has redshifted to ' + str(zCMBfuture) + ', the radius of the CMB shell will be ' + str(round(d,1)) + ' kpc larger'

tdiff_Myr = cd.lookback_time(zCMBfuture,zCMBnow,**cosmo) / cc.Myr_s
print 'This will take ' + str(round(tdiff_Myr*1e6)) + ' years.'