  Galaxies obey a number of empirical correlations between their radio,
{\gamma}-ray, and infrared emission, but the physical origins of these
correlations remain uncertain. Here we use the CONGRuENTS model for broadband
non-thermal emission from star-forming galaxies, which self-consistently
calculates energy-dependent transport and non-thermal emission from cosmic ray
hadrons and leptons, to predict radio and {\gamma}-ray emission for a synthetic
galaxy population with properties drawn from a large deep-field survey. We show
that our synthetic galaxies reproduce observed relations such as the FIR-radio
correlation, the FIR-{\gamma} correlation, and the distribution of radio
spectral indices, and we use the model to explain the physical origins of these
relations. Our results show that the FIR-radio correlation arises because the
amount of cosmic ray electron power ultimately radiated as synchrotron emission
varies only weakly with galaxy star formation rate as a result of the
constraints imposed on gas properties by hydrostatic balance and turbulent
dynamo action; the same physics dictates the extent of proton calorimetry in
different galaxies, and thus sets the FIR-{\gamma}-ray correlation. We further
show that galactic radio spectral indices result primarily from competition
between thermal free-free emission and energy-dependent loss of cosmic ray
electrons to bremsstrahlung and escape into galactic halos, with shaping of the
spectrum by inverse Compton, synchrotron, and ionisation processes typically
playing a sub-dominant role. In addition to explaining existing observations,
we use our analysis to predict a heretofore unseen correlation between the
curvature of galaxies' radio spectra and their pion-driven {\gamma}-ray
emission, a prediction that will be testable with upcoming facilities.
