The interaction between the wall and the core region of turbulent channels is studied using direct numerical simulations at friction Reynolds number ${\hbox{\it Re}}_{\tau} \approx 630$. In these simulations the near-wall energy cycle is effectively removed, replacing the smooth-walled boundary conditions by prescribed velocity disturbances with non-zero Reynolds stress at the walls. The profiles of the first- and second-order moments of the velocity are similar to those over rough surfaces, and the effect of the boundary condition on the mean velocity profile is described using the equivalent sand roughness. Other effects of the disturbances on the flow are essentially limited to a layer near the wall whose height is proportional to a length scale defined in terms of the additional Reynolds stress. The spectra in this roughness sublayer are dominated by the wavenumber of the velocity disturbances and by its harmonics. The wall forcing extracts energy from the flow, while the normal equilibrium between turbulent energy production and dissipation is restored in the overlap region. It is shown that the structure and the dynamics of the turbulence outside the roughness sublayer remain virtually unchanged, regardless of the nature of the wall. The detached eddies of the core region only depend on the mean shear, which is not modified beyond the roughness sublayer by the wall disturbances. On the other hand, the large scales that are correlated across the whole channel scale with $U_{\hbox{\scriptsize{\it LOG}}}=u_{\tau}\kappa^{-1}\log({\hbox{\it Re}}_\tau)$, both in smooth- and in rough-walled flows. This velocity scale can be interpreted as a measure of the velocity difference across the log layer, and it is used to modify the scaling proposed and validated by del Álamo et al. (J. Fluid Mech., vol. 500, 2004, p. 135) for smooth-walled flows.