  We present the first detailed study of the kinematics of free relativistic
particles whose symmetries are described by a quantum deformation of the de
Sitter algebra, known as $q$-de Sitter Hopf algebra. The quantum deformation
parameter is a function of the Planck length $\ell$ and the de Sitter radius
$H^{-1}$, such that when the Planck length vanishes, the algebra reduces to the
de Sitter algebra, while when the de Sitter radius is sent to infinity one
recovers the $\kappa$-Poincar\'e Hopf algebra. In the first limit the picture
is that of a particle with trivial momentum space geometry moving on de Sitter
spacetime, in the second one the picture is that of a particle with de Sitter
momentum space geometry moving on Minkowski spacetime. When both the Planck
length and the inverse of the de Sitter radius are non-zero, effects due to
spacetime curvature and non-trivial momentum space geometry are both present
and affect each other. The particles' motion is then described in a full phase
space picture. We find that redshift effects that are usually associated to
spacetime curvature become energy-dependent. Also, the energy dependence of
particles' travel times that is usually associated to momentum space
non-trivial properties is modified in a curvature-dependent way.
