  We investigate the quantum motion of a neutral Dirac particle bouncing on a
mirror in curved spacetime. We consider different geometries: Rindler,
Kasner-Taub and Schwarzschild, and show how to solve the Dirac equation by
using geometrical methods. We discuss, in a first-quantized framework, the
implementation of appropriate boundary conditions. This leads us to consider a
Robin boundary condition that gives the quantization of the energy, the
existence of bound states and of critical heights at which the Dirac particle
bounces, extending the well-known results established from the Schrodinger
equation. We also allow for a nonminimal coupling to a weak magnetic field. The
problem is solved in an analytical way on the Rindler spacetime. In the other
cases, we compute the energy spectrum up to the first relativistic corrections,
exhibiting the contributions brought by both the geometry and the spin. These
calculations are done in two different ways. On the one hand, using a
relativistic expansion and, on the other hand, with Foldy-Wouthuysen
transformations. Contrary to what is sometimes claimed in the literature, both
methods are in agreement, as expected. Finally, we make contact with the GRANIT
experiment. Relativistic effects and effects that go beyond the equivalence
principle escape the sensitivity of such an experiment. However, we show that
the influence of a weak magnetic field could lead to observable phenomena.
