Observations of evolution in the dust-to-metal ratio allow to constrain the dominant dust processing mechanisms. In this work, we present a study of the dust-to-metal and dust-to-gas ratios in a subsample of ~500 DustPedia galaxies. Using literature and MUSE emission line fluxes, we derive gas-phase metallicities (oxygen abundances) for over 10000 individual regions and determine characteristic metallicities for each galaxy. We study how the relative dust, gas and metal contents of galaxies evolve by using metallicity and gas fraction as proxies for evolutionary state. The global oxygen abundance and nitrogen-to-oxygen ratio are found to increase monotonically as galaxies evolve. Additionally, unevolved galaxies (gas fraction >60%, metallicity 12+log(O/H)<8.2) have dust-to-metal ratios that are about a factor of 2.1 lower (factor of 6 lower for galaxies with gas fraction >80%) than the typical dust-to-metal ratio (Md/MZ~0.214) for more evolved sources. However, for high gas fractions, the scatter is larger due to larger observational uncertainties as well as a potential dependence of the dust grain growth timescale and supernova dust yield on local conditions and star formation histories. We find chemical evolution models with a strong contribution from dust grain growth describe these observations reasonably well. The dust-to-metal ratio is also found to be lower for low stellar masses and high specific star formation rates (with the exception of some sources undergoing a starburst). Finally, the metallicity gradient correlates weakly with the HI-to-stellar mass ratio, the effective radius and the dust-to-stellar mass ratio, but not with stellar mass.