Not necessarily, if they can produce the crops more cheaply. Since each country ideally wants to secure it's own food supply, it's inevitable that many countries will find themselves subsidizing local production that would otherwise disappear in a competitive international market.

Additionally, hostile countries do not need to flood markets sustainably if the goal is simply to hollow out food production in the target country before taking more overtly hostile (i.e. military) actions.

Dump and pump?