  Much progress has been made recently in the acceleration of $\sim10^{4}$\,K
clouds to explain absorption-line measurements of the circumgalactic medium and
the warm, atomic phase of galactic winds. However, the origin of the cold,
molecular phase in galactic winds has received relatively little theoretical
attention. Studies of the survival of $\sim10^{4}$\,K clouds suggest efficient
radiative cooling may enable the survival of expelled material from galactic
disks. Alternatively, gas colder than 10$^4$\,K may form within the outflow,
including molecules if dust survives the acceleration process. We explore the
survival of dusty clouds in a hot wind with three-dimensional hydrodynamic
simulations including radiative cooling and dust modeled as tracer particles.
We find that cold $\sim10^{3}$\,K gas can be destroyed, survive, or transformed
entirely to $\sim 10^4\,$K gas. We establish analytic criteria distinguishing
these three outcomes which compare characteristic cooling times to the system's
`cloud crushing' time. In contrast to typically studied $\sim10^{4}$\,K clouds,
colder clouds are entrained faster than the drag time as a result of efficient
mixing. We find that while dust can in principle survive embedded in the
accelerated clouds, the survival fraction depends critically on the time dust
spends in the hot phase and on the effective threshold temperature for
destruction. We discuss our results in the context of polluting the
circumgalactic medium with dust and metals, as well as understanding
observations suggesting rapid acceleration of molecular galactic winds and ram
pressure stripped tails of jellyfish galaxies.
