We study stellar and black hole mass assembly in a sample of 42 infrared-luminous galaxy mergers at z < 0.3 by combining results from radiative transfer modelling with archival measures of molecular gas and black hole mass. The ratios of stellar mass, molecular gas mass, and black hole mass to each other are consistent with those of massive gas-rich galaxies at z < 0.3. The advanced mergers may show increased black hole mass to stellar mass ratios, consistent with the transition from AGN to ellipticals and implying substantial black hole mass growth over the course of the merger. Star formation rates are enhanced relative to the local main sequence, by factors of ∼100 in the starburst and ∼1.8 in the host, respectively. The starburst star formation rates appear distinct to star formation in the main sequence at all redshifts up to at least z ∼ 5. Starbursts may prefer late-stage mergers, but are observed at any merger stage. We do not find evidence that the starbursts in these low-redshift systems substantially increase the total stellar mass, with a soft upper limit on the stellar mass increase from starburst activity of about a factor of two. In contrast, 12 objects show evidence for super-Eddington accretion, associated with late-stage mergers, suggesting that many AGN in infrared-luminous mergers go through a super-Eddington phase. The super-Eddington phase may increase black hole mass by up to an order of magnitude at an accretion efficiency of $42\pm 33{{\ \rm per\ cent}}$ over a period of 44 ± 22 Myr. Our results imply that super-Eddington accretion is an important black hole growth channel in infrared-luminous galaxies at all redshifts.