Topological defects in ferroic materials are attracting much attention both as a playground of unique physical phenomena and for potential applications in reconfigurable electronic devices. Here, we explore electronic transport at artificially created ferroelectric vortices in BiFeO3 thin films. The creation of one-dimensional conductive channels activated at voltages as low as 1V is demonstrated. We study the electronic as well as the static and dynamic polarization structure of several topological defects using a combination of first-principles and phase-field modelling. The modelling predicts that the core structure can undergo a reversible transformation into a metastable twist structure, extending charged domain walls segments through the film thickness. The vortex core is therefore a dynamic conductor controlled by the coupled response of polarization and electron-mobile-vacancy subsystems with external bias. This controlled creation of conductive one-dimensional channels suggests a pathway for the design and implementation of integrated oxide electronic devices based on domain patterning.