Abstract We continue our comprehensive theoretical investigation of the spectral evolution of white dwarfs based on sophisticated simulations of element transport. In this paper, we focus on the transformation of PG 1159 stars into DO/DB white dwarfs due to the gravitational settling of heavy elements and then into DQ white dwarfs through the convective dredge-up of carbon. We study the impact of several physical parameters on the evolution of the surface carbon abundance over a wide range of effective temperatures. In the hot PG 1159 and DO phases, our calculations confirm that the temperature of the PG 1159-to-DO transition depends sensitively on the stellar mass and the wind mass-loss rate. We show that measured carbon abundances of DOZ white dwarfs are mostly accounted for by our models, with the notable exception of the coolest DOZ stars. In the cooler DB and DQ phases, the predicted atmospheric composition is strongly influenced by the stellar mass, the thickness of the envelope, the initial carbon content, the efficiency of convective overshoot, and the presence of residual hydrogen. We demonstrate that, under reasonable assumptions, our simulations reproduce very well the observed carbon abundance pattern of DQ stars, which thus allows us to constrain the extent of the overshoot region in cool helium-rich white dwarfs. We also argue that our calculations naturally explain a number of recent empirical results, such as the relative excess of low-mass DQ stars and the presence of trace hydrogen and/or carbon at the surface of most DC and DZ stars.