In this study, we propose that the curvature of graphene can be exploited to perform directional molecular motion and provide atomistic insights into the curvature-dependent molecular migration through density functional theory calculations. We first reveal the origin of the different migration trends observed experimentally for aromatic molecules with electron-donating and -withdrawing groups on p-doped functionalized graphene. Next, we show that the kinetic barrier for migration depends on the amount and nature of the curvature, that is, positive versus negative curvature. We find that the molecular migration on a wrinkled/rippled graphene sheet preferentially happens from the valley (positive curvature) to the mountain (negative curvature) regions. To understand the origin of such curvature-dependent molecular motion and migrational kinetic barrier trends, we develop a descriptor based on the frontier orbital orientation of graphene. Finally, based on these findings, we predict that time- and space-varying curvature can drive directional molecular motion on graphene and thus further propose that efforts should focus on exploring other two-dimensional materials as active platforms for performing controlled molecular motion.