Intersections of sumsets in additive number theory

Let $A$ be a subset of an additive abelian semigroup $S$ and let $hA$ be the $h$-fold sumset of $A$. The following question is considered: Let $(A_q)_{q=1}^{\infty}$ be a strictly decreasing sequence of sets in $S$ and let $A = \bigcap_{q=1}^{\infty} A_q$. When does one have $$hA = \bigcap_{q=1}^{\infty} hA_q$$ for some or all $h \geq 2$?