We present a study of a sample of 254 clusters from the SDSS-DR7 Yang Catalogue and an auxiliary sample of field galaxies to perform a detailed investigation on how galaxy quenching depends on both environment and galaxy stellar mass. Our samples are restricted to 0.03 ≤ z ≤ 0.1 and we only consider clusters with log (Mhalo/M⊙) ≥ 14. Comparing properties of field and cluster galaxies in the blue cloud, green valley, and red sequence, we find evidence that field galaxies in the red sequence hosted star formation events $\rm 2.1 \pm 0.7$ Gyr ago, on average, more recently than galaxies in cluster environments. Dissecting the star formation rate versus stellar mass diagram we show how morphology rapidly changes after reaching the green valley region, while the star formation rate keeps decreasing. In addition, we use the relation between location in the projected phase space and infall time to explore the time delay between morphological and specific star formation rate variations. We estimate that the transition from late- to early-type morphology happens in Δtinf ~ 1 Gyr, whereas the quenching of star formation takes ~3 Gyr. The time-scale we estimate for morphological transitions is similar to the expected for the delayed-then-rapid quenching model. Therefore, we suggest that the delay phase is characterized mostly by morphological transition, which then contributes morphological quenching as an additional ingredient in galaxy evolution.