The dynamics of a vortex subject to a localized stretching is numerically investigated. The structure of the flow is analysed in the case of an initially two-dimensional vortex surrounded by a periodic array of vortex rings localized far from its core. Amplified oscillations of both the axial vorticity and the stretching are found, in strong contrast with Burgers-like vortices. The resulting dynamics is the appearance, around the vortex, of successive vortical structures of smaller and smaller radius and alternate sign embedded in the previous vortical rings. The frequency scaling of the oscillations is recovered by linear analysis (Kelvin modes) but not the amplification nor the shape of the successive tori. An inviscid model based on structures is presented, which compares better with the numerical computations. These results suggest that the formalism of Kelvin waves is not sufficient to describe the full dynamics, which is instead related to the feedback of rotation on stretching and more conveniently described in terms of localized structures. We finally discuss the relative timescales of vortex stretching and of vortex reaction. The Burgers-like vortices, where there is no such reaction, turn out to correspond to a nearly pure strain field, slightly disturbed by rotation.