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The most active phases of star formation and black hole accretion are strongly affected by dust extinction, making far-infrared (FIR) 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 and ionised and neutral atomic and warm molecular gas in the rest-frame mid-infrared (MIR) and FIR provide unmatched diagnostic opportunities 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 FIR spectroscopy, the PRobe far-Infrared Mission for Astrophysics (PRIMA), a 1.8 m diameter, cryogenically cooled FIR observatory currently at the conception stage, 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 SPRITZand 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 10¹¹ L_⊙beyond cosmic noon and at least up toz = 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 hydrocarbons (PAHs), and the total IR luminosity. At the same time, spectroscopic follow up with PRIMA will allow us to trace both the star formation and black hole accretion rates (SFRs and BHARs), the gas-phase metallicities, and the mass-outflow rates of cold gas in hundreds to thousands of individual galaxies toz = 2.