We derive sufficient conditions on a drift vector field to let an asymptotically stable invariant submanifold of an input-affine system without drift remain asymptotically stable for the system with drift. In doing so, we use the same feedback laws modulo control gain tuning, such that no new feedback laws need to be designed for the system with drift. Our main assumption is that the vector field of the input-affine system without drift assumes the form of a gradient vector field for given feedback laws. We show how one can assess the performance of the system with drift only via knowledge about the system without drift. Finally, we find that our results are relevant in synchronization problems and backstepping controllers for mechanical systems.