  The most active phases of star formation and black hole accretion are
strongly affected by dust extinction, making far-infrared (far-IR) observations
the best way to disentangle and study the co-evolution of galaxies and super
massive black holes. The plethora of fine structure lines and emission features
from dust, ionised and neutral atomic and warm molecular gas in the rest-frame
mid- and far-IR provide unmatched diagnostic power to determine the properties
of gas and dust, measure gas-phase metallicities and map cold galactic outflows
in even the most obscured galaxies. By combining multi-band photometric surveys
with low and high-resolution far-IR spectroscopy, the PRobe far-Infrared
Mission for Astrophysics (PRIMA), a concept for a far-IR, 1.8m-diameter,
cryogenically cooled observatory, will revolutionise the field of galaxy
evolution by taking advantage of this IR toolkit to find and study dusty
galaxies across galactic time. In this work, we make use of the
phenomenological simulation SPRITZ and the Santa Cruz semi-analytical model to
describe how a moderately deep multi-band PRIMA photometric survey can easily
reach beyond previous IR missions to detect and study galaxies down to
$10^{11}\,L_{\odot}$ beyond cosmic noon and at least up to z=4, even in the
absence of gravitational lensing. By decomposing the spectral energy
distribution (SED) of these photometrically selected galaxies, we show that
PRIMA can be used to accurately measure the relative AGN power, the mass
fraction contributed by polycyclic aromatic hydrocarbon (PAH) and the total IR
luminosity. At the same time, spectroscopic follow up with PRIMA will allow to
trace both the star formation and black hole accretion rates (SFR, BHAR), the
gas phase metallicities and the mass outflow rates of cold gas in hundreds to
thousands of individual galaxies to z=2.
