  The geodesic equation encodes test-particle dynamics at arbitrary
gravitational coupling, hence retaining all orders in the post-Minkowskian (PM)
expansion. Here we explore what geodesic motion can tell us about dynamical
scattering in the presence of perturbatively small effects such as tidal
distortion and higher derivative corrections to general relativity. We derive
an algebraic map between the perturbed geodesic equation and the leading PM
scattering amplitude at arbitrary mass ratio. As examples, we compute formulas
for amplitudes and isotropic gauge Hamiltonians for certain infinite classes of
tidal operators such as electric or magnetic Weyl to any power, and for higher
derivative corrections to gravitationally interacting bodies with or without
electric charge. Finally, we present a general method for calculating
closed-form expressions for amplitudes and isotropic gauge Hamiltonians in the
test-particle limit at all orders in the PM expansion.
