The effects of mobile vacancies on the dynamics of ordering processes and phase separation in multicomponent systems are studied via Monte Carlo simulations of a two-dimensional seven-state ferromagnetic Potts model with varying degrees of site dilution. The model displays phase equilibria corresponding to a dilute Potts-disordered (fluid) phase and a dilute Potts-ordered phase (solid), as well as a broad region of coexistence between the fluid and the solid phase. Temperature quenches into the dilute Potts-ordered phase as well as into the phase-separated region are considered under the condition of conserved vacancy density and nonconserved Potts order. The dynamics of ordering and phase separation is found to follow algebraic growth laws with exponent values that depend on the phase to which the quench is performed. Strong transient effects are observed in the dilute Potts-ordered phase, which are shown to result from an accumulation of vacancies at the domain boundaries. These transient effects are accompanied by an overshooting effect in the local order of the growing domains.