  For a surjective self-morphism on a projective variety defined over a number
field, we study the preimages question, which asks if the set of rational
points on the iterated preimages of an invariant closed subscheme eventually
stabilize. We prove the answer is positive for self-morphisms on $\mathbb{P}^1
\times \mathbb{P}^1$. We also prove that the answer is positive even if we
allow finite extension of the number field by a bounded degree if the morphism
is \'etale or the invariant subscheme is $0$-dimensional. We prove certain
uniformity of the stabilization for self-morphisms on abelian varieties. When
the subscheme is not invariant, we provide counterexamples and a result in a
positive direction for products of polynomial maps on $\mathbb{P}^1 \times
\mathbb{P}^1$.
