Droplet actuation on wires or fibers is highly relevant to digital microfluidics applications. Several different passive or active actuation mechanisms have been studied, including capillary pressure, thermal gradient, tilting, acoustics, and electrowetting, to name a few. However, the lack of precise control and accurate droplet positioning during its actuation on a wire limits the use of earlier methods. On the other hand, using a nonuniform magnetic field to actuate a droplet on a micrometer-size wire helps overcome the above limitations. In this context, we elucidate the motion of microliter-size ferrofluid droplets, smaller than the capillary length, in the presence of a permanent magnet using a simple yet robust experimental setup, wherein the clamshell shape droplets hang on a wire at a predetermined distance from the magnet. Beyond a critical volume, the droplet starts moving toward the magnet while exhibiting shape evolution continuously. Using a high-speed videography technique, we uncover the intricate relationship between the magnetowetting, favoring continuous shape evolution of droplets of different sizes, and their actuation dynamics (velocity) on a wire, which has scarcely been explored. The most significant contribution of this study is the detailed evaluation of the shape factor (k), relating the shape evolution of the droplet to its velocity through the contact angle hysteresis force. A theoretical framework has also been proposed, comprising different forces in action, which is found to set forth a good match with the experimental results. The results from this study are expected to open up new avenues in digital microfluidics, specifically in drug delivery, ferrobotics, and droplet logic gates, to name a few.