Galaxies arrive on the red sequences of clusters at high redshift (z>1) once their star formation is quenched and evolve passively thereafter. However, here we have found that cluster red sequence galaxies (CRSGs) undergo significant morphological evolution subsequent to the cessation of star formation, at some point in the past 9-10~Gyr. Through a detailed study of a large sample of cluster red sequence galaxies spanning 0.2 < z < 1.4 we elucidate the details of this evolution. Below z ~ 0.5-0.6 (in the last 5-6 Gyr) there is little or no morphological evolution in the population as a whole, unlike in the previous 4-5 Gyrs. Over this earlier time (i) disk-like systems with Sersic n < 2 progressively disappear, at the same tims as (ii) the range of their axial ratios similarly decreases, removing the most elongated systems (those consistent with thin disks seen at an appreciable inclination angle), and (iii) radial colour gradients (bluer outwards) decrease in an absolute sense from significant age-related gradients to a residual level consistent with the metallicity-induced gradients seen in low redshift cluster members. The distribution of their effective radii shows some evidence of evolution, consistent with growth of {\it at most} a factor <1.5 between z~ 1.4 and z ~ 0.5, significantly less than for comparable field galaxies at least at z<1. A simple model involving the fading and thickening of a disk component after comparatively recent quenching (after z~ 1.5) around an otherwise passively evolving older spheroid component is moderately consistent with all of these findings.