Controlling magnetization dynamics is imperative for developing ultrafast spintronics and tunable microwave devices. However, the previous research has demonstrated limited electric-field modulation of the effective magnetic damping, a parameter that governs the magnetization dynamics. Here, we propose an approach to manipulate the damping by using the large damping enhancement induced by the two-magnon scattering and a nonlocal spin relaxation process in which spin currents are resonantly transported from antiferromagnetic domains to ferromagnetic matrix in a mixed-phased metallic alloy FeRh. This damping enhancement in FeRh is sensitive to its fraction of antiferromagnetic and ferromagnetic phases, which can be dynamically tuned by electric fields through a strain-mediated magnetoelectric coupling. In a heterostructure of FeRh and piezoelectric PMN-PT, we demonstrated a more than 120% modulation of the effective damping by electric fields during the antiferromagnetic-to-ferromagnetic phase transition. Our results demonstrate an efficient approach to controlling the magnetization dynamics, thus enabling low-power tunable electronics.