We investigate the correlation between far-infrared (FIR) and radio luminosities in distant galaxies, a lynchpin of modern astronomy. We use data from the Balloon-borne Large Aperture Submillimetre Telescope (BLAST), Spitzer, the Large Apex BOlometer CamerA (LABOCA), the Very Large Array and the Giant Metre-wave Radio Telescope (GMRT) in the Extended Chandra Deep Field South (ECDFS). For a catalogue of BLAST 250-mu m-selected galaxies, we remeasure the 70-870-mu m flux densities at the positions of their most likely 24-mu m counterparts, which have a median [interquartile] redshift of 0.74 [0.25, 1.57]. From these, we determine the monochromatic flux density ratio, q(250)(= log(10) [S-250 mu m/S-1400 MHz]), and the bolometric equivalent, q(IR). At z approximate to 0.6, where our 250-mu m filter probes rest-frame 160-mu m emission, we find no evolution relative to q(160) for local galaxies. We also stack the FIR and submm images at the positions of 24-mu m- and radio-selected galaxies. The difference between q(IR) seen for 250-mu m- and radio-selected galaxies suggests that star formation provides most of the IR luminosity in less than or similar to 100-mu Jy radio galaxies, but rather less for those in the mJy regime. For the 24-mu m sample, the radio spectral index is constant across 0 < z < 3, but q(IR) exhibits tentative evidence of a steady decline such that q(IR) proportional to (1 + z)-0.15 +/- 0.03 - significant evolution, spanning the epoch of galaxy formation, with major implications for techniques that rely on the FIR/radio correlation. We compare with model predictions and speculate that we may be seeing the increase in radio activity that gives rise to the radio background.