ABSTRACT Recent years have seen excellent progress in modelling the entrainment of T ∼ 104 K atomic gas in galactic winds. However, the entrainment of cool, dusty T ∼ 10–100 K molecular gas, which is also observed outflowing at high velocity, is much less understood. Such gas, which can be 105 times denser than the hot wind, appears extremely difficult to entrain. We run 3D wind-tunnel simulations with photoionization self-shielding and evolve thermal dust sputtering and growth. Unlike almost all such simulations to date, we do not enforce any artificial temperature floor. We find efficient molecular gas formation and entrainment, as well as dust survival and growth through accretion. Key to this success is the formation of large amounts of 104 K atomic gas via mixing, which acts as a protective ‘bubble wrap’ and reduces the cloud overdensity to χ ∼ 100. This can be understood from the ratio of the mixing to cooling time. Before entrainment, when shear is large, tmix/tcool ≲ 1, and gas cannot cool below the ‘cooling bottleneck’ at 5000 K. Thus, the cloud survival criterion is identical to the well-studied purely atomic case. After entrainment, when shear falls, tmix/tcool > 1, and the cloud becomes multiphase, with comparable molecular and atomic masses. The broad temperature PDF, with abundant gas in the formally unstable $50 \, {\rm K} \lt T \lt 5000 \, {\rm K}$ range, agrees with previous ISM simulations with driven turbulence and radiative cooling. Our findings have implications for dusty molecular gas in stellar and active galactic nuclei outflows, cluster filaments, ‘jellyfish’ galaxies, and asymptomatic giant branch winds.