In this work, we used a generalized Frenkel-Kontorova model to study the mobility of water molecules inside carbon nanotubes with small radius at low temperatures. Our simulations show that the mobility of confined water decreases monotonically increasing the amplitude of the substrate potential at fixed commensurations. On the other hand, the mobility of the water molecules shows a non-monotonic behavior when varying the commensuration. This result indicates that the mobility of the confined fluid presents different behavior regimes depending on the amplitude of the water-nanotube interaction. In order to qualitatively understand these results, we study analytically the driven Frenkel-Kontorova model at finite temperatures. This analysis allows us to obtain the curves of the mobility versus commensurations, at fixed substrate potentials. Such curves show the existence of three regimes of mobility behavior as a function of the commensuration ratio. Additionally, our study indicates a nontrivial and strong dependence of the mobility with a quantity that can be interpreted as an effective amplitude of the substrate potential, depending on the bare amplitude of the substrate potential, the commensuration ratio, and temperature.