  In this paper, we provide the first computations for the distortion transfer
functions of the cosmic microwave background (CMB) in the perturbed Universe,
following up on paper I and II in this series. We illustrate the physical
effects inherent to the solutions, discussing and demonstrating various
limiting cases for the perturbed photon spectrum. We clarify the relationship
between distortion transfer functions and the photon spectrum itself, providing
the machinery that can then compute constrainable CMB signal power spectra
including spectral distortions for single energy injection and decaying
particle scenarios. Our results show that the $\mu \times T$ and $y\times T$
power spectra reach levels that can be constrained with current and future CMB
experiments without violating existing constraints from COBE/FIRAS. The
amplitude of the cross-correlation signal directly depends on the average
distortion level, therefore establishing a novel fundamental link between the
state of the primordial plasma from redshift $10^3 \lesssim z\lesssim 3\times
10^6$ and the frequency-dependent CMB sky. This provides a new method to
constrain average early energy release using CMB imagers. As an example we
derive constraints on single energy release and decaying particle scenarios.
This shows that LiteBIRD may be able to improve the energy release limits of
COBE/FIRAS by up to a factor of $\simeq 2.5$, while PICO could tighten the
constraints by more than one order of magnitude. The signals considered here
could furthermore provide a significant challenge to reaching cosmic
variance-limited constraints on primordial non-Gaussianity from distortion
anisotropy studies. Our work further highlights the immense potential for a
synergistic spectroscopic approach to future CMB measurements and analyses.
