We consider non-stationary oscillations of an infinite string with time-varying tension. The string lies on the Winkler foundation with a point inhomogeneity (a concentrated spring of negative stiffness). In such a system with constant parameters (the string tension), under certain conditions a trapped mode of oscillation exists and is unique. Therefore, applying a non-stationary external excitation to this system can lead to the emergence of the string oscillations localized near the inhomogeneity. We provide an analytical description of non-stationary localized oscillations of the string with slowly time-varying tension using the asymptotic procedure based on successive application of two asymptotic methods, namely the method of stationary phase and the method of multiple scales. The obtained analytical results were verified by independent numerical calculations based on the finite difference method. The applicability of the analytical formulas was demonstrated for various types of external excitation and laws governing the varying tension. In particular, we have shown that in the case when the trapped mode frequency approaches zero, localized low-frequency oscillations with increasing amplitude precede the localized string buckling. The dependence of the amplitude of such oscillations on its frequency is more complicated in comparison with the case of a one-degree-of-freedom system with time-varying stiffness.