The faceting of a growing crystal is theoretically investigated by a continuum model including the incorporation kinetics of adatoms. This allows us for predictions beyond a simple Wulff analysis which typically refers to faceted morphologies in terms of the equilibrium crystal shape for crystals with an anisotropic surface-energy, or to steady-state kinetic shape when the crystals grow with orientation-dependent velocities. A phase-field approach is implemented in order to account simultaneously for these contributions in two- and three dimensions reproducing realistic kinetic pathways for the morphological evolution of crystal surfaces during growth. After a systematic characterization of the faceting determined by orientation-dependent incorporation times, several different crystal morphologies are found by tuning the relative weights of thermodynamic and kinetic driving forces. Applications to realistic systems are finally reported showing the versatility of the proposed approach and demonstrating the key role played by the incorporation dynamics in out-of-equilibrium growth processes.