Axisymmetric vortex core flows, in unconfined and confined geometries, are examined using a quasi-one-dimensional analysis. The goal is to provide a simple unified view of the topic which gives insight into the key physical features, and the overall parametric dependence, of the core area evolution due to boundary geometry or far-field pressure variation. The analysis yields conditions under which waves on vortex cores propagate only downstream (supercritical flow) or both upstream and downstream (subcritical flow), delineates the conditions for a Kelvin–Helmholtz instability arising from the difference in core and outer flow axial velocities, and illustrates the basic mechanism for suppression of this instability due to the presence of swirl. Analytic solutions are derived for steady smoothly, varying vortex cores in unconfined geometries with specified far-field pressure and in confined flows with specified bounding area variation. For unconfined vortex cores, a maximum far-field pressure rise exists above which the vortex cannot remain smoothly varying; this coincides with locally critical conditions (axial velocity equal to wave speed) in terms of wave propagation. Comparison with axisymmetric Navier–Stokes simulations and experimental results indicate that this maximum correlates with the appearance of vortex breakdown and marked core area increase in the simulations and experiments. For confined flows, the core stagnation pressure defect relative to the outer flow is found to be the dominant factor in determining conditions for large increases in core size. Comparisons with axisymmetric Navier–Stokes computations show that the analysis captures qualitatively, and in many instances, quantitatively, the evolution of vortex cores in confined geometries. Finally, a strong analogy with quasi-one-dimensional compressible flow is demonstrated by construction of continuous and discontinuous flows as a function of imposed downstream core edge pressure.