We derive dust masses ($M_{rm dust}$) from the spectral energy distributions of 58 post-starburst galaxies (PSBs). There is an anticorrelation between specific dust mass ($M_{rm dust}$/$M_{star}$) and the time elapsed since the starburst ended, indicating that dust was either destroyed, expelled, or rendered undetectable over the $sim$1 Gyr after the burst. The $M_{rm dust}$/$M_{star}$ depletion timescale, 205$^{+58}_{-37}$ Myr, is consistent with that of the CO-traced $M_{rm H_2}/M_{star}$, suggesting that dust and gas are altered via the same process. Extrapolating these trends leads to the $M_{rm dust}/M_{star}$ and $M_{rm H_2}/M_{star}$ values of early-type galaxies (ETGs) within 1-2 Gyr, a timescale consistent with the evolution of other PSB properties into ETGs. Comparing $M_{rm dust}$ and $M_{rm H_2}$ for PSBs yields a calibration, log $M_{rm H_2}$ = 0.45 log $M_{rm dust}$ + 6.02, that allows us to place 33 PSBs on the Kennicutt-Schmidt (KS) plane, $Sigma rm SFR-Sigma M_{rm H_2}$. Over the first $sim$200-300 Myr, the PSBs evolve down and off of the KS relation, as their star formation rate (SFR) decreases more rapidly than $M_{rm H_2}$. Afterwards, $M_{rm H_2}$ continues to decline whereas the SFR levels off. These trends suggest that the star-formation efficiency bottoms out at 10$^{-11} rm yr^{-1}$ and will rise to ETG levels within 0.5-1.1 Gyr afterwards. The SFR decline after the burst is likely due to the absence of gas denser than the CO-traced H$_2$. The mechanism of the $M_{rm dust}/M_{star}$ and$M_{rm H_2}/M_{star}$ decline, whose timescale suggests active galactic nucleus (AGN) or low-ionization nuclear emission-line region (LINER) feedback, may also be preventing the large CO-traced molecular gas reservoirs from collapsing and forming denser star forming clouds.
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