Current-carrying string-loop dynamics is studied in the Kerr spacetimes. With attention concentrated to the axisymmetric motion of string loops around the symmetry axis of both black-hole (BH) and naked singularity (NS) spacetimes, it is shown that the resulting motion is governed by the presence of an outer tension barrier and an inner angular momentum barrier that are influenced by the BH or NS spin. We classify the string dynamics according to properties of the energy boundary function (effective potential) for the string loop motion. We have found that for NS there exist new types of energy boundary function, namely those with off-equatorial minima. Conversion of the energy of the string oscillations to the energy of the linear translational motionhas been studied. Such a transmutation effect is much more efficient in the NS spacetimes because of lack of the event horizon. For BH spacetimes efficiency of the transmutation effect is only weakly spin dependent. Transition from the r