Direct Numerical Simulations (DNS) of Couette–Poiseuille turbulent channel flows have been performed to understand whether universality of the flow near the stationary and the moving wall occurs. As in previous experiments, we have observed that the flow near the stationary wall is not affected by the flow in the rest of the channel. Near the moving wall streaky structures form if the normalized shear rate Sk/ε is larger than a threshold value, which is found to agree with that found in other turbulent flows. The statistics in the region near the moving wall are not observed to scale with the friction velocity, and a better collapse of the data is instead obtained in terms of dissipative units evaluated at the wall.