In many astrophysical systems, mixing between cool and hot temperature gas/plasma through Kelvin-Helmholtz-instability-driven turbulence leads to the formation of an intermediate temperature phase with increased radiative losses that drive efficient cooling. The solar atmosphere is a potential site for this process to occur with interaction between either prominence or spicule material and the solar corona allowing the development of transition region material with enhanced radiative losses. In this paper, we derive a set of equations to model the evolution of such a mixing layer and make predictions for the mixing-driven cooling rate and the rate at which mixing can lead to the condensation of the coronal material. These theoretical predictions are benchmarked against 2.5D MHD simulations. Applying the theoretical scalings to prominence threads or fading spicules, we found that as a mixing layer grows on their boundaries this would lead to the creation of transition region material with a cooling time of ~100 s, explaining the warm emission observed as prominence threads or spicules fade in cool spectral lines without the requirement for any heating. For quiescent prominences, dynamic condensation driven by the mixing process could restore ~18 per cent of the mass lost from a prominence through downflows. Overall, this mechanism of thermal energy loss through radiative losses induced by mixing highlights the importance for considering dynamical interaction between material at different temperatures when trying to understand the thermodynamic evolution of the cool material in the solar corona.