The stability properties of magnetic flux tubes in stellar convection zones including overshoot regions is of considerable interest in connection with the problems of magnetic flux storage and hydromagnetic dynamo action in the Sun and other cool stars. We have developed a general formalism based on the approximation of thin flux tubes which provides a basis for a linear stability analysis of arbitrary flux tube equilibria. As a first application, the stability of axisymmetric, toroidal flux tubes (flux rings) located in the equatorial plane of a star under the influence of differential rotation and stratification has been considered. Arbitrary angular velocity differences between the interior of the flux ring and its environment are permitted. It is found that the linear evolution of radial and azimuthal perturbations (i.e., within the equatorial plane) is decoupled from that of latitudinal perturbations (perpendicular to the plane). The latitudinal instability (‘poleward slip’) is found to be suppressed if the matter within the flux tube rotates faster than its environment by a sufficient amount. For perturbations within the equatorial plane, both stratification (sub-order superadiabatic) of the external gas and rotation are crucial. Angular momentum conservation tends to suppress axisymmetric modes. This effect is enhanced by a faster rotation of the gas within the flux tube. Non-axisymmetric modes are more unstable since the constraint of angular momentum conservation is broken. For these modes, a slower internal rotation rate has a stabilizing effect. Within a certain range of magnetic field strengths, a second region of stability exists within the region of unstable configurations, which can extend into the superadiabatically stratified (convectively unstable) region. The character of the different modes is discussed in conjunction with the topology of the stability diagram.