We study a three-dimensional holographic conformal field theory under the influence of a background electric field on a spacetime containing two black hole horizons. The electric background is fixed such that there is potential difference between the two boundary black holes, inducing a conserved current. By constructing the holographic duals to this setup, which are solutions to the Einstein-Maxwell equations with a negative cosmological constant in four dimensions, we calculate, to a fully nonlinear level, the conductivity of the conformal field theory in this background. Interestingly, we find that the conductivity depends nontrivially on the potential difference. The bulk solutions are flowing geometries containing black hole horizons which are non-Killing and have nonzero expansion. We find a novel property that the past boundary of the future horizon lies deep in the bulk and show this property remains present after small perturbations of the temperature difference of the boundary black holes. Published by the American Physical Society 2024