Phononic crystals and elastic metamaterials have recently received significant attention due to their potential for unconventional wave control. Despite this interest, one outstanding issue is that their band diagram is typically fixed once the structure is designed. To overcome this limitation, periodic structures with adaptive elastic properties have recently been proposed for Bragg- and local resonance-driven structures. In this context, we report about the effect of an applied external mechanical load on a periodic structure exhibiting band gaps induced by inertial amplification mechanism. If compared to the cases of Bragg scattering and ordinary local resonant metamaterials, we observe here a more remarkable curve shift, modulated through large but fully reversible compression (stretch) of the unit cell, eventually triggering significant (up to two times) enlargement (reduction) of the width of a specific band gap. An important up (down)-shift of some dispersion branches over specific wavenumber values is also observed, showing that this selective variation may lead to negative group velocities over larger (smaller) wavenumber ranges. In addition, the possibility for a non-monotone trend of the lower limit of the first BG under the same type of external applied prestrain is found and explained through an analytical model, which unequivocally proves that this behaviour derives from the different unit cell effective mass and stiffness variations as the prestrain level increases. These peculiarities derive from the hinge-like behaviour of some regions of the unit cell, which is typical of structures exhibiting the inertial amplification mechanism. The effect of the prestress on the dispersion diagram is investigated through the development of a 2-step calculation method: first, an Updated Lagrangian scheme, including a static geometrically nonlinear analysis of a representative unit cell undergoing the action of an applied external load is derived, and then the Floquet-Bloch decomposition is applied to the linearized equations of the acousto-elasticity for the unit cell in the deformed configuration. Finally, the most evident consequence on the dispersion curves of the application of an external prestress, i.e. the band gap shift with respect to the unloaded structure, is demonstrated through nonlinear transient numerical simulations, clearly proving the capability of the structure to switch from a pass- to a stop-band behaviour over the same frequency range. The results presented herein provide insights in the behaviour of band gaps induced by inertial amplification, and suggest new opportunities for real-time tunable wave manipulation.