The Lagrangian equations of motion for massive spinning test particles (tops) moving on a gravitational background using general relativity are presented. The paths followed by tops are nongeodesic.An exact solution for themotion of tops on a Schwarzschild background which allows for superluminal propagation of tops is studied. It is shown that the solution becomes relevant for particles with small masses, such as neutrinos. This general result is used to calculate the necessary condition to produce superluminal motion in part of the trajectory of a small mass particle in a weak gravitational field. The condition for superluminal motion establishes a relation between the mass, energy and total angular momentum of the particle.