This paper presents a numerical study of the dynamics and stability of two-dimensional thermal plumes in a significantly stratified layer. Motivated by stellar envelope convection in which radiative cooling at the star's photosphere drives vigorous down flows, we examine cool plumes descending through an adiabatically stratified layer of increasing density with depth. Such flows are inaccessible by laboratory experiments, yet are important to the understanding of heat and momentum transport, magnetic field generation, and acoustic excitation in stars like the Sun. We find that the structure of thermal plumes in a stratified compressible medium is significantly different from that in an incompressible one, with pressure perturbations playing an important dynamical role. Additionally, we find that the plumes are subject to vigorous secondary instabilities even in a quiescent background medium. While the flows studied are not fully turbulent but transitional, the nature of the compressive instabilities and their influence on subsequent flow evolution suggests that advective detrainment of fluid from the plume region results. Simplified plume models assuming a hydrostatic pressure distribution and velocity-proportional entrainment may thus be inappropriate in this context.