  Quantifying changes in galaxies' interstellar medium (ISM) abundance after
quenching star formation is an important aspect of galaxy evolution, but it is
poorly constrained beyond the local universe. We characterise the dust-related
properties in 548 quiescent galaxies observed at $0.1<z<0.6$ as part of the
hCOSMOS spectroscopic survey. This is the largest sample of quiescent galaxies
at intermediate redshifts, for which the co-evolution of dust, metals and stars
have been estimated. We reveal the complex relations between the key markers of
galaxies' dust life-cycles, such as specific dust mass ($M_{\rm dust}$/$M_{\rm
\star}$), with gas-metallicity ($Z_{\rm gas}$), time since quenching ($t_{\rm
quench}$), stellar age and size. We find morphology to be important factor of a
large scatter ($\sim2$ orders of magnitude) in $M_{\rm dust}/M_{\rm \star}$.
Through modelling the star formation histories of our objects, we derive a
broad dynamical range of post-quenching timescales ($60\:\rm Myr<t_{\rm
quench}<3.2\:\rm Gyr$). We find that $M_{\rm dust}/M_{\rm \star}$ is the
highest in recently quenched systems ($t_{\rm quench}<500$ Myr), but its
further evolution is non-monotonic as a consequence of diverse pathways for
prolonged dust formation, or removal on various timescales. Our data are well
reproduced by the SIMBA cosmological simulation and chemical models that
include dust growth in the ISM. While this process is prevalent in dusty
quiescent galaxies, $\sim15\%$ of objects show signs of external dust
acquisition, most likely via minor mergers. Our results strongly suggest that
prolonged dust production on a timescale $0.5-1\:\rm Gyr$ since quenching may
be common in dusty quiescent galaxies at intermediate redshifts, even if their
gas reservoirs are heavily exhausted (i.e., cold gas fraction $<1-5\%$).
