We simulate gravity-driven dynamics of partially wetting droplets moving along a fiber using the lattice Boltzmann method. For the so-called clamshell morphology, we find three possible dynamic regimes upon varying the droplet Bond number and the fiber radius: compact droplet, droplet breakup, and droplet oscillation. For small Bond numbers, in the compact droplet regime, the droplet reaches a steady state and its velocity scales linearly with the driving body force. We find two scaling laws depending on whether the droplet size is smaller or larger compared to the fiber radius. In addition, we further identify a scaling law applicable for the barrel morphology. For higher Bond numbers, in the droplet breakup regime, satellite droplets are formed trailing the initial moving droplet. We find such droplet satellite formation is easier with increasing fiber curvature (smaller fiber radius). Finally, in the droplet oscillation regime, favored in the midrange of fiber radius, the droplet shape periodically extends and contracts along the fiber.