Generally, a topos over a base topos is called an atomic topos if is atomic.
See (Johnstone, p. 689).
Atomic morphisms are closed under composition.
An atomic geometric morphism is also a locally connected geometric morphism.
This appears as (Johnstone, lemma 3.5.4).
This appears as (Johnstone, cor. 3.5.2).
If logical then it preserves the isomorphism characterizing a Boolean topos and hence is Boolean if it is atomic.
For the converse…
This appears as (Johnstone, lemma 3.5.4 (iii)).
Every etale geometric morphism is atomic.
Section C3.5 of