We study the mobility of extended objects (rods) on a spherical liquid-liquid interface to show how this quantity is modified in a striking manner by both the curvature and the topology of the interface. We present theoretical calculations and experimental measurements of the interfacial fluid velocity field around a moving rod bound to the crowded interface of a water-in-oil droplet. By using different droplet sizes, membrane viscosities, and rod lengths, we show that the viscosity mismatch between the interior and exterior fluids leads to a suppression of the fluid flow on small droplets that cannot be captured by the flat interface predictions. ; Comment: 4 pages, 3 figures