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Aims. We study the dust content of massive (log(M*/M⊙)≥10.8) quiescent galaxies (QGs) at redshifts z = 0.5 − 3 to place constraints on the evolution of their cold interstellar medium (ISM) and thereby obtain insights into the processes of galaxy quenching throughout cosmic time. Methods. We used a robust sample of 458 colour-selected QGs covered by the A3COSMOS+A3GOODSS database to perform a stacking analysis in the uv domain and measured their mean dust masses from their stacked sub-millimetre luminosities. We used the <ns0:monospace>CIGALE</ns0:monospace> spectral energy distribution fitting code to obtain star formation histories and infer the time since quenching for all the QGs in our sample. We used this information to gain insight into the time evolution of the dust content after quenching. Results. Most QGs in our sample quenched around a redshift of z ∼ 1.3, following the peak of cosmic star formation. The majority of QGs observed at z > 1 are recently quenched (i.e. quenched for no longer than 500 Myr), whereas the majority of QGs observed at z < 1 have already been quenched for a significant amount of time (≳1 Gyr). This implies that high-redshift galaxies (z ≳ 2) are ideal for studying the mechanisms of quenching and its effects on the ISM, while lower-redshift galaxies are more suitable for studying the long-term effects of the QG environment on their ISM. We obtain upper limits on the dust mass fraction of the QG population that indicate a lower dust content in high-redshift massive QGs than what was found by earlier stacking studies, and significantly lower (by a factor of ∼2–6) than that of normal star-forming galaxies. We also place constraints on the initial gas fraction right after quenching. We find that within the first ∼600 Myr after quenching, QGs already lose on average ≳70% of their cold ISM. Our findings support a gas consumption or removal scenario acting on short timescales.