The current-driven domain wall motion along two exchange-coupled ferromagnetic layers with perpendicular anisotropy is studied by means of micromagnetic simulations and compared to the conventional case of a single ferromagnetic layer. Our results, where only the lower ferromagnetic layer is subjected to the interfacial Dzyaloshinskii-Moriya interaction and to the spin Hall effect, indicate that the domain walls can be synchronously driven in the presence of a strong interlayer exchange coupling, and that the velocity is significantly enhanced due to the antiferromagnetic exchange coupling as compared with the single-layer case. On the contrary, when the coupling is of ferromagnetic nature, the velocity is reduced. We provide a full micromagnetic characterization of the current-driven motion in these multilayers, both in the absence and in the presence of longitudinal fields, and the results are explained based on a one-dimensional model. The interfacial Dzyaloshinskii-Moriya interaction, only necessary in this lower layer, gives the required chirality to the magnetization textures, while the interlayer exchange coupling favors the synchronous movement of the coupled walls by a dragging mechanism, without significant tilting of the domain wall plane. Finally, the domain wall dynamics along curved strips is also evaluated. These results indicate that the antiferromagnetic coupling between the ferromagnetic layers mitigates the tilting of the walls, which suggest these systems to achieve efficient and highly packed displacement of trains of walls for spintronics devices. A study, taking into account defects and thermal fluctuations, allows to analyze the validity range of these claims.