In prior work we found that precise approximation of the continuity constraint is crucial for accurate propagation of tracer data when advected through a background incompressible velocity field (Sime et al., 2021). Here we extend this investigation to compressible flows using the anelastic liquid approximation (ALA) and address four related issues: 1. exact conservation of tracer discretized fields through a background compressible velocity; 2. exact mass conservation; 3. addition and removal of tracers without affecting (exact) conservation to preserve a consistent number of tracers per cell; and 4. the diffusion of tracer data, for example, as induced by thermal or chemical effects. In this process we also present an abstract formulation of the interior penalty hybrid discontinuous Galerkin (HDG) finite element formulation for diffusion problems, and apply it to the advection-diffusion and compressible Stokes systems. Finally we present numerical experiments exhibiting the HDG compressible Stokes momentum formulation’s superconvergent compressibility approximation and reproduce community numerical benchmarks from the literature for the ALA.