Abstract We analyse the molecular and atomic emission lines of 71 bright Herschel-selected galaxies between redshifts 1.4 to 4.6 detected by the Atacama Large Millimetre/submillimetre Array. These lines include a total of 156 CO, [C i], and H2O emission lines. For 46 galaxies, we detect two transitions of CO lines, and for these galaxies we find gas properties similar to those of other dusty star-forming galaxy (DSFG) samples. A comparison to photo-dissociation models suggests that most of Herschel-selected galaxies have similar interstellar medium conditions as local infrared-luminous galaxies and high-redshift DSFGs, although with denser gas and more intense far-ultraviolet radiation fields than normal star-forming galaxies. The line luminosities agree with the luminosity scaling relations across five orders of magnitude, although the star-formation and gas surface density distributions (i.e., Schmidt-Kennicutt relation) suggest a different star-formation phase in our galaxies (and other DSFGs) compared to local and low-redshift gas-rich, normal star-forming systems. The gas-to-dust ratios of these galaxies are similar to Milky Way values, with no apparent redshift evolution. Four of 46 sources appear to have CO line ratios in excess of the expected maximum (thermalized) profile, suggesting a rare phase in the evolution of DSFGs. Finally, we create a deep stacked spectrum over a wide rest-frame frequency (220–890 GHz) that reveals faint transitions from HCN and CH, in line with previous stacking experiments.