In this paper, we prove a priori estimates for some vortex-type equations on compact Riemann surfaces. As applications, we recover existing estimates for the vortex bundle Monge-Ampère equation, and prove an existence and uniqueness theorem for the Calabi-Yang-Mills equations on vortex bundles. We also prove an existence and uniqueness result relating Gieseker stability and the existence of almost Hermitian Einstein metrics, i.e., a Kobayashi-Hitchin type correspondence.