Much progress has been made recently in the acceleration of ∼ 10 4 K clouds to explain absorption-line measurements of the circumgalactic medium and the atomic phase of galactic winds. However, the origin of the molecular phase in galactic winds has received relatively little theoretical attention. Studies of the survival of atomic clouds suggest efficient radiative cooling may enable the survival of expelled material from galactic disks. Alternatively, atomic and molecular gas may form within the outflow, if dust survives the acceleration process. We explore the survival of molecular, dusty clouds in a hot wind with three-dimensional hydrodynamic simulations in which we include radiative cooling and model dust as tracer particles. We find that molecular gas can be destroyed, survive, or transformed entirely to ∼ 10 4 K gas. We establish analytic criteria distinguishing these three outcomes which compare characteristic cooling times to the 'cloud crushing' time of the system. In contrast to typically studied atomic ∼ 10 4 K clouds, molecular clouds are entrained faster than the drag time as a result of efficient mixing. Moreover, 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.