  In this paper, we prove that given a flat generically smooth morphism between
smooth projective varieties with $F$-pure closed fibers, if the source space is
Fano, weak Fano or a variety with the nef anti-canonical divisor, then so is
the target space. We also show that relative anti-canonical divisors of
generically smooth surjective nonconstant morphisms are not nef and big in
arbitrary characteristic.
