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Antiferromagnetic insulators present a promising alternative to ferromagnets due to their ultrafast spin dynamics essential for low-energy terahertz spintronic device applications. Magnons, i.e., quantized spin waves capable of transmitting information through excitations, serve as a key functional element in this paradigm. However, identifying external mechanisms to effectively tune magnon properties has remained a major challenge. Here we demonstrate that interfacial metal-insulator transitions offer an effective method for controlling the magnons of Sr2IrO4, a strongly spin-orbit coupled antiferromagnetic Mott insulator. Resonant inelastic x-ray scattering experiments reveal a significant softening of zone-boundary magnon energies in Sr2IrO4 films epitaxially interfaced with metallic 4d transition-metal oxides. Therefore, the magnon dispersion of Sr2IrO4 can be tuned by metal-insulator transitions of the 4d transition-metal oxides. We tentatively attribute this non-trivial behavior to a long-range phenomenon mediated by magnon-acoustic phonon interactions. Our experimental findings introduce a strategy for controlling magnons and underscore the need for further theoretical studies to better understand the underlying microscopic interactions between magnons and phonons.