We consider the far-from-equilibrium quantum transport dynamics in a 1D Josephson junction chain of multimode Bose-Einstein condensates. We develop a theoretical model to examine the experiment of Labouvie et al. [Phys. Rev. Lett. 115, 050601 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.050601], wherein the phenomenon of negative differential conductivity (NDC) was reported in the refilling dynamics of an initially depleted site within the chain. We demonstrate that a unitary c-field description can quantitatively reproduce the experimental results over the full range of tunnel couplings, and requires no fitted parameters. With a view toward atomtronic implementations, we further demonstrate that the filling is strongly dependent on spatial phase variations stemming from quantum fluctuations. Our findings suggest that the interpretation of the device in terms of NDC is invalid outside of the weak coupling regime. Within this restricted regime, the device exhibits a hybrid behavior of NDC and the ac Josephson effect. A simplified circuit model of the device will require an approach tailored to atomtronics that incorporates quantum fluctuations.