  We define affine transport lifts on the tangent bundle by associating a
transport rule for tangent vectors with a vector field on the base manifold.
The aim is to develop tools for the study of kinetic/ dynamical symmetries in
relativistic particle motion. The transport lift unifies and generalises the
various existing lifted vector fields, with clear geometric interpretations. We
find the affine dynamical symmetries of free particle motion, and compare this
to previous results and to the alternative concept of "matter symmetry".
