Carbon nanotubes (CNT) are known to provide a hydrophobic, confined environment for water where its structure and dynamics can be very different from those of bulk water. In particular, narrow CNTs of the type (6,6) allow only a single one-dimensional (1D) chain of water molecules inside them, thus providing an idealized scenario to study motion in ID along water chains. In the present study, we have investigated structural and dynamic behavior of water and also of an excess proton and hydroxide ion in water-filled narrow CNTs by means of ab initio molecular dynamics and combined quantum-classical simulations. The main focus of the present work is on the molecular mechanism and kinetics of hydronium and hydroxide ion migration along ID water chains of different lengths in confinement. It is found that the hydrogen-bonded structures of water and the excess proton and hydroxide ion in CNTs can be very different from those in bulk, and these altered solvation structures play critical roles in determining the proton-transfer (PT) rates along water chains. For the present ID chain systems, the hydroxide ion is found to migrate at a slightly faster rate than the excess proton, unlike their relative mobilities in bulk water. This faster migration of the hydroxide ion is found not only in CNTs with periodicity along the tube axis but also in isolated CNTs where the excess proton and the hydroxide ion are allowed to move under the influence of an electric field of an oppositely charged ion. The roles of rotational jumps and hydrogen-bond fluctuations in the PT events are discussed. In addition, the significance of hydrogen-bonding defects on the dynamics of an excess proton and hydroxide ion is also discussed for varying chain lengths.