The Particle Strength Exchange (PSE) scheme is useful for modeling diffusion of vorticity in a purely Lagrangian numerical construction. This is done in a fully-coupled non-split implementation of diffusion and convection. Unfortunately, there is no similarly general Lagrangian scheme for modeling the arbitrary circulation source terms, such as those due to baroclinic vorticity generation in a reacting flow environment. We present and analyze both non-split and split hybrid Lagrangian-Eulerian formulations that allow the modeling of baroclinic vorticity generation. We find that the non-split implementation, even though fully coupled, does not achieve second-order time accuracy in the context of a second-order Runge-Kutta construction. On the other hand, the symmetric operator-split formulation does achieve second-order convergence, and is clearly superior in this context. This formulation is being presently used in the context of hybrid Lagrangian-Eulerian computations of reacting flow.