  Arrays of coupled photonic cavities driven by external lasers represent a
highly controllable setup to explore photonic transport. In this paper we
address (quasi)-steady states of this system that exhibit photonic currents
introduced by engineering driving and dissipation. We investigate two
approaches: in the first one, photonic currents arise as a consequence of a
phase difference of applied lasers and in the second one, photons are injected
locally and currents develop as they redistribute over the lattice. Effects of
interactions are taken into account within a mean-field framework. In the first
approach, we find that the current exhibits a resonant behavior with respect to
the driving frequency. Weak interactions shift the resonant frequency toward
higher values, while in the strongly interacting regime in our mean-field
treatment the effect stems from multiphotonic resonances of a single driven
cavity. For the second approach, we show that the overall lattice current can
be controlled by incorporating few cavities with stronger dissipation rates
into the system. These cavities serve as sinks for photonic currents and their
effect is maximal at the onset of quantum Zeno dynamics.
