  We present the latest version of the GAEA model of galaxy formation. Our new
model combines (i) an updated treatment of AGN feedback including an improved
modelling of cold gas accretion on super-massive BHs and an explicit
implementation of quasar winds; and (ii) an improved modelling of cold and hot
gas stripping from satellite galaxies. We show that our latest model predicts
specific SFR distributions that are in remarkable agreement with observational
measurements in the local Universe. Our updated model predicts quenched
fractions that are in very nice agreement with data up to z~3-4, and a
turn-over of the number densities of quenched galaxies at low stellar masses
that is in qualitative agreement with current observational estimates. We show
that the main reasons for the improved behaviour with respect to previous
renditions of our model are the updated treatment for satellites at low galaxy
masses (<10^10 Msun) and the inclusion of quasar winds at intermediate to large
stellar masses (>10^10 Msun). However, we show that the better treatment of the
star formation threshold, due to our explicit partitioning of the cold gas in
its atomic and molecular components, also plays an important role in
suppressing excessive residual star formation in massive galaxies. While our
analysis is based on a selection of quiescent galaxies that takes advantage of
the information about their SFR, we demonstrate that the impact of a different
(colour-colour) selection is not significant, at least for galaxies above the
completeness limits of current surveys. Our new model predicts number densities
of massive quiescent galaxies at z>3 that are the largest among recently
published models. Yet, our model predictions appear still to be below post-JWST
observational measurements. We show that the expected cosmic variance is large,
and can easily accommodate some of the most recent measurements.
