ABSTRACT We investigate the evolution in colour and morphology of the progenitors of red-sequence galaxies in the EAGLE cosmological hydrodynamical simulation. We quantify colours with u* − r* intrinsic magnitudes and morphologies with a measure of the stellar kinematics. The time when galaxies moved on to the red sequence depends on their morphology. Disc-type galaxies tend to have become red during the last 3 Gyr, while elliptical-type galaxies joined the red sequence earlier, with half the sample already being red 5 Gyr ago. The time-scale, τGreen, of colour transition through the ‘green valley’ depends weakly on the galaxy’s morphological type. Elliptical-type galaxies cross the green valley slightly faster (τGreen ≈ 1 Gyr) than disc-type galaxies (τGreen ≈ 1.5 Gyr). While τGreen is similar for central and satellite galaxies, for satellites τGreen decreases with increasing stellar mass to host-halo mass ratio. Coupled with our finding that galaxies tend to become green after becoming satellites, this indicates that satellite-specific processes are important for quenching red-sequence galaxies. The last time central, elliptical-type red-sequence galaxies left the blue cloud is strongly correlated with the time the luminosity of the central black hole peaked, but this is not the case for discs. This suggests that active galactic nuclei (AGNs) feedback is important for quenching ellipticals, particularly centrals, but not for discs. We find only a weak connection between transformations in colour and morphology.