We present results from nonequilibrium molecular dynamics experiments describing the relaxation of local concentrations at two reservoirs, initially filled with water (W) and acetonitrile (ACN), as they become connected through a membrane composed of (16,16) carbon nanotubes. Within the hydrophobic nanotube cavities, the equilibrium concentrations contrast sharply to those observed at the reservoirs, with a clear enhancement of ACN, in detriment of W. From the dynamical side, the relaxation involves three well-differentiated stages; the first one corresponds to the equilibration of individual concentrations within the nanotubes. An intermediate interval with Fickian characteristics follows, during which the overall transport can be cast in terms of coaxial opposite fluxes, with a central water domain segregated from an external ACN shell, in close contact with the tube walls. We also found evidence of a third, much slower, mechanism to reach equilibration, which involves structural modifications of tightly bound solvation shells, in close contact with the nanotube rims.