We discuss relaxation and aging processes in the one- and two-dimensional ABC models. In these driven diffusive systems of three particle types, biased exchanges in one direction yield a coarsening process characterized in the long-time limit by a logarithmic growth of ordered domains that take the form of stripes. From the time-dependent length, derived from the equal-time spatial correlator, and from the mean displacement of individual particles different regimes in the formation and growth of these domains can be identified. Analysis of two-times correlation and response functions reveals dynamical scaling in the asymptotic logarithmic growth regime as well as complicated finite-time and finite-size effects in the early and intermediate time regimes.