  We investigate the soft behavior of the tree-level Rutherford scattering
processes mediated via $t$-channel one-graviton exchange. We consider two types
of Rutherford scattering processes, {\it e.g.}, a low-energy massless
structureless projectile (up to spin-$1$) hits a static massive composite
particle carrying various spins (up to spin-$2$), and a slowly-moving light
projectile hits a heavy static composite target. The unpolarized cross sections
in the first type are found to exhibit universal forms at the first two orders
in $1/M$ expansion, yet differ at the next-to-next-to-leading order, though
some terms at this order still remain universal or depend on the target spin in
a definite manner. The unpolarized cross sections in the second type are
universal at the lowest order in projectile velocity expansion and through all
orders in $1/M$, independent of the spins of both projectile and target. The
universality partially breaks down at relative order-$v^2/M^2$, albeit some
terms at this order still depend on the target spin in a specific manner.
