Let F : W --> V be a dominant rational map between quasi-projective varieties
of the same dimension. We give two proofs that h_V(F(P)) >> h_W(P) for all
points P in a nonempty Zariski open subset of W. For dominant rational maps F :
P^n --> P^n, we give a uniform estimate in which the implied constant depends
only on n and the degree of F. As an application, we prove a specialization
theorem for equidimensional dominant rational maps to semiabelian varieties,
providing a complement to Habegger's recent theorem on unlikely intersections.