Antiferromagnetic insulators offer an alternative to ferromagnets due to their ultrafast spin dynamics essential for low-energy terahertz spintronic device applications. One way is to utilize magnons, i.e., quantized spin waves, which can carry information through excitations. However, finding external knobs for tuning the magnons has been a significant challenge. Here we report that interfacial metal-insulator transitions can be an effective means for controlling the magnons of a strongly spin-orbit-coupled antiferromagnetic Mott insulator, Sr2IrO4. From resonant inelastic X-ray scattering and Raman spectroscopy, we have observed a pronounced softening of zone-boundary magnon energies in several Sr2IrO4 thin-film systems that are epitaxially contacted with metallic 4d transition-metal oxides (TMOs). Therefore, the magnon dispersion of Sr2IrO4 is tunable by metal-insulator transitions of the 4d TMO crystals. Remarkably, this non-trivial behavior of magnons is a long-range phenomenon coupled with intriguing magnon-phonon interactions. Our experimental finding proposes a new scheme for magnonics.