First, avoid unwise solution:
Global food shortages is not merely a crisis of one nation, but crisis of humankind. And humans should not be deprived of such basic necessity. Any government that prioritizes only its own citizens during a global crisis shows not only how immature they are, but also how inhumane and hypocritical they can be.
If governments are NOT practicing according to Jesus' creed, "love your neighbors as yourselves", then they shall have no rights to advocate, and should not expect their own citizens to cultivate such practice within their own communities. Thus, helping neighboring countries during this crisis cannot stop. The government has to be a moral example too. And as soon as they realize this, this time of crisis can well be a time of opportunity.
Second, on possible solution:
In a regional crisis of food, the region should diversify and enlarge the choices of staple food instead of sticking to only one: rice/wheat. Governments should encourage and educate their own nations to consume other types of food. Therefore governments should source for other edible resources to their nation, and not limiting their effort to source only for one or two types of staple food. In fact limiting sourcing choices is a main reason that causes food panic. These are the leasts that any government can do. Banning neighboring citizens from buying food is probably not the most feasible and exemplary way.