The entry of solar wind into the magnetosphere is strongly influenced by kinetic-scale boundary layers where the rapid variation in the magnetic field and/or velocity can drive transport. In current layers with strong Alfvénic velocity shear, the generation of vortices from the Kelvin-Helmholtz instability can drive magnetic reconnection even in broader current sheets by locally compressing these layers as the vortices develop. Previous two-dimensional (2-D) fully kinetic simulations of this vortex-induced reconnection process have demonstrated the copious formation of magnetic islands in regions of strongly compressed current between the vortices. Here we describe the first three-dimensional (3-D) fully kinetic simulations of this process and demonstrate that the compressed current sheets give rise to magnetic flux ropes over a range of oblique angles and along the entire extent of the compressed current layer around the periphery of the vortex. These flux ropes propagate with the shear flow and eventually merge with the vortex. Over longer time scales, this basic scenario is repeated as the vortices drive new compressed current sheets. In the final stage, the vortices undergo a merging process that drives new compressed current sheets and flux ropes. Based on these simulations, a simple model is proposed that predicts the size of these flux ropes relative to their parent vortex. Both the relative sizes as well as the structure of the profiles across the vortex are in reasonable agreement with Time History of Events and Macroscale Interactions (THEMIS) observations at the Earth's low-latitude magnetopause.