Time-accurate direct numerical simulations with detailed chemistry and transport were used to study transient effects during the transitions between different flame structures in an axisymmetric hydrogen/air opposed-jet burner. By starting from a steady flame and impulsively changing the flowrates, we simulated two different scenarios: (1) extinction of diffusion flames around the axis of symmetry and formation of a propagating edge flame that ultimately leads to a ring-shaped edge flame stabilizing away from the axis of symmetry, and (2) propagation of ring-shaped premixed flames along the stoichiometric isoline through a weakly stratified mixture in the direction normal to the direction of propagation, eventually closing the flame hole and re-establishing a disk-shaped diffusion flame. A front moving to quench or extend the diffusion flames is observed at the initial and final stages of cases 1 and 2, respectively. After extinction of or before re-establishment of the diffusion tail, the flames (as a whole) move in a quasi-steady manner with a propagating velocity relative to the fresh reactants that is close to the premixed laminar flame speed. No negatively propagating edge flame was found during extinction; before the transition to a strongly burning edge flame, the diffusion flame edge is convected outward with the local flow velocity. During re-ignition, the edge flame propagates all the way to the axis, and no homogeneous re-ignition was observed. Close to the axis it attains a propagation velocity that is almost twice the laminar flame speed.