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Precipitation changes in a warming climate have been examined with a focus on either mean precipitation or precipitation extremes, but changes in the full probability distribution of precipitation have not been well studied. This paper develops a methodology for the quantile-conditional column moisture budget of the atmosphere for the full probability distribution of precipitation. Analysis is performed on idealized aquaplanet model simulations under 3-K uniform SST warming across different horizontal resolutions. Because the covariance of specific humidity and horizontal mass convergence is much reduced when conditioned onto a given precipitation percentile range, their conditional averages yield a clear separation between the moisture (thermodynamic) and circulation (dynamic) effects of vertical moisture transport on precipitation. The thermodynamic response to idealized climate warming can be understood as a generalized "wet get wetter" mechanism, in which the heaviest precipitation of the probability distribution is enhanced most from increased gross moisture stratification, at a rate controlled by the change in lower-tropospheric moisture rather than column moisture. The dynamic effect, in contrast, can be interpreted by shifts in large-scale atmospheric circulations such as the Hadley cell circulation or midlatitude storm tracks. Furthermore, horizontal moisture advection, albeit of secondary role, is important for regional precipitation change. Although similar mechanisms are at play for changes in both mean precipitation and precipitation extremes, the thermodynamic contributions of moisture transport to increases in high percentiles of precipitation tend to be more widespread across a wide range of latitudes than increases in the mean, especially in the subtropics.