The nucleation and dynamics of Josephson and Abrikosov vortices determine the critical currents of layered high-Tc superconducting (HTS) thin films, grain boundaries, and coated conductors, so understanding their mechanisms is of crucial importance. Here, we treat pair creation of Josephson and Abrikosov vortices in layered superconductors as a secondary Josephson effect. Each full vortex is viewed as a composite fluid of micro-vortices, such as pancake vortices, which tunnel coherently via a tunneling matrix element. We introduce a two-terminal magnetic (Weber) blockade effect that blocks tunneling when the applied current is below a threshold value. We simulate vortex tunneling as a dynamic, time-correlated process when the current is above threshold. The model shows nearly precise agreement with voltage-current (V-I) characteristics of HTS cuprate grain boundary junctions, which become more concave rounded as temperature decreases, and also explains the piecewise linear V-I behavior observed in iron-pnictide bicrystal junctions and other HTS devices. When applied to either Abrikosov or Josephson pair creation, the model explains a plateau seen in plots of critical current vs. thickness of HTS-coated conductors. The observed correlation between theory and experiment strongly supports the proposed quantum picture of vortex nucleation and dynamics in layered superconductors.