By means of numerical simulations, we demonstrate that an alternating-current (ac) field can support stably moving collective nonlinear excitations in the form of dislocations (topological solitons, or kinks) in the Frenkel-Kontorova (FK) lattice with weak friction, as was qualitatively predicted by Bonilla and Malomed (Bonilla L L and Malomed B A 1991 Phys. Rev. B 43 11 539). Direct generation of the moving dislocations turns out to be virtually impossible; however, they can be generated initially in the lattice subject to an auxiliary spatial modulation of the on-site potential strength. Gradually relaxing the modulation, we are able to get stable moving dislocations in the uniform FK lattice with periodic boundary conditions, provided that the driving frequency is close to the gap frequency of the linear excitations in the uniform lattice. The excitations that can be generated in this way have a large and noninteger index of commensurability with the lattice (so suggesting that the actual value of the commensurability index is irrational). The simulations reveal two different types of moving dislocation: broad ones, that extend, roughly, to half the full length of the periodic lattice (in that sense, they cannot be called solitons); and localized soliton-like dislocations, that can be found in an excited state, demonstrating strong persistent internal vibrations. The minimum (threshold) amplitude of the driving force necessary to support the travelling excitation is found as a function of the friction coefficient. Its extrapolation suggests that the threshold does not vanish at zero friction, which may be explained by radiation losses. The moving dislocation can be observed experimentally in an array of coupled small Josephson junctions in the form of an inverse Josephson effect, i.e., a direct-current-voltage response to the uniformly applied ac bias current.