We study the dynamics of particles evolving in a two-dimensional periodic, spatially-symmetric potential landscape. The system is subjected to weak external time-periodic forces rocking the potential in either direction which, inter alia, breaks integrability. In particular, chaotic layers arising around separatrices which connect unstable equilibria constitute a network of interlinked paths along which trajectories can diffuse. Attention is paid to the emergence of directed motion for an ensemble of particles being trapped initially inside one well of the two-dimensional potential landscape. Interestingly, with properly chosen values of the strength, frequencies, and phases, of the external modulation forces, the particle motion can not only be initiated but also targeted in a desired direction. We demonstrate that, although unstable motion in the form of Arnold diffusion is possible, non-biased diffusion of an ensemble of particles is restricted to certain regions in phase space such that the resulting net motion proceeds directedly. This gives evidence that higher-dimensional tori play the role of partial barriers impeding unhindered motion in phase space and, in addition, serve to capture trajectories in transporting channels.