  We show that if the automorphism group of a projective variety is torsion,
then it is finite. Motivated by Lang's conjecture on rational points of
hyperbolic varieties, we use this to prove that a projective variety with only
finitely many rational points has only finitely many automorphisms. Moreover,
we investigate to what extent finiteness of $S$-integral points on a variety
over a number field persists over finitely generated fields. To this end, we
introduce the class of mildly bounded varieties and prove a general criterion
for proving this persistence.
