Methods. We have modelled a sample of ~800 nearby galaxies, spanning a wide range of metallicity, gas fraction, specific star formation rate and Hubble stage. We have derived the dust properties of each object from its spectral energy distribution. Through an additional level of analysis, we have inferred the timescales of dust condensation in core-collapse supernova ejecta, grain growth in cold clouds and dust destruction by shock waves. Throughout this paper, we have adopted a hierarchical Bayesian approach, resulting in a single large probability distribution of all the parameters of all the galaxies, to ensure the most rigorous interpretation of our data. Results. We confirm the drastic evolution with metallicity of the dust-to-metal mass ratio (by two orders of magnitude), found by previous studies. We show that dust production by core-collapse supernovae is efficient only at very low-metallicity, a single supernova producing on average less than ~0.03 Msun/SN of dust. Our data indicate that grain growth is the dominant formation mechanism at metallicity above ~1/5 solar, with a grain growth timescale shorter than ~50 Myr at solar metallicity. Shock destruction is relatively efficient, a single supernova clearing dust on average in at least ~1200 Msun/SN of gas. These results are robust when assuming different stellar initial mass functions. In addition, we show that early-type galaxies are outliers in several scaling relations. This feature could result from grain thermal sputtering in hot X-ray emitting gas, an hypothesis supported by a negative correlation between the dust-to-stellar mass ratio and the X-ray photon rate per grain. Finally, we confirm the well-known evolution of the aromatic-feature-emitting grain mass fraction as a function of metallicity and interstellar radiation field intensity. Our data indicate the relation with metallicity is significantly stronger.
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