  We examine the spheroid growth and star formation quenching experienced by
galaxies from z~3 to the present by studying the evolution with redshift of the
quiescent and spheroid-dominated fractions of galaxies from the CANDELS and
GAMA surveys. We compare the observed fractions with predictions from a
semi-analytic model which includes prescriptions for bulge growth and AGN
feedback due to mergers and disk instabilities. We facilitate direct
morphological comparison by converting our model bulge-to-total stellar mass
ratios to Sersic indices. We then subdivide our population into the four
quadrants of the sSFR-Sersic index plane and study the buildup of each of these
subpopulations. We find that the fraction of star forming disks declines
steadily, while the fraction of quiescent spheroids builds up over cosmic time.
The fractions of star forming spheroids and quiescent disks are both
non-negligible, and stay nearly constant over the period we have studied, at
about 10% and 15-20% respectively. Our model is qualitatively successful at
reproducing the evolution of the two "main" populations (star forming
disk-dominated galaxies and quiescent spheroid-dominated galaxies), and
approximately reproduces the relative fractions of all four types, but predicts
a stronger decline in star forming spheroids, and increase in quiescent disks,
than seen in the observations. A model with an additional channel for bulge
growth via disk instabilities agrees better overall with the observations than
a model in which bulges may grow only through mergers. We study evolutionary
tracks of some individual galaxies as they experience morphological
transformation and quenching, and examine the importance of different physical
drivers of this transformation (major and minor mergers and disk
instabilities). We find that complex histories with multiple transformative
events are the norm.
