Compound viscous jets composed of an inner core of one fluid surrounded by an annulus of another are studied here using long-wave theory. We investigate the limit of strongly differing viscosities in the neighbouring fluids: either a highly viscous core or annulus. Additionally, if inertia is present, the density ratio of the fluids is assumed to mirror that of the viscosities. The resulting asymptotic theory reduces to several earlier theories in the appropriate limits. Linear stability analysis and full numerical simulations of the one-dimensional set of equations allow an exploration of the dynamics of either the interior or exterior fluid. The results of our linear stability analysis demonstrate that, for both cases, a stretching and a squeezing mode exist, with the former being more dominant than the latter. Our numerical simulations show that in the highly viscous core case, the interfaces can move out-of-phase, leading to the breakup of the annular region; this is contrary to the linear theory predictions. In the highly viscous annulus case, our results demonstrate the possibility of breakup of either the core or the annulus, depending on the initial ratio of the radii.