Pinning and plastic flow of superconducting phases of current-driven Josephson junction lattices in magnetic fields are studied using numerical simulations. We consider two kinds of lattice geometry: a two-dimensional array (Josephson junction array, JJA) and a ladder (Josephson junction ladder, JJL), and introduce positional disorder to the systems. In these systems in the presence of driving dc currents, the plastic deformation of a vortex lattice or the nucleation of vortex-antivortex pairs causes pinned, plastic flow and elastic flow states. We focus on two threshold currents that characterize the boundaries of these regimes. We analyse the threshold currents on the basis of the scaling theory of pinning due to random potentials. It is found that, for both JJL and JJA, the threshold currents show scaling behaviours characterized by certain scaling exponents. We discuss the relationship between the threshold currents and the parameters of the junctions. We also discuss the nonlinear dynamics of plastic flow and the transition from plastic to elastic flow of the present systems in comparison with other related models.