Prices go up because demand exceeds supply. The higher prices of such crops will induce the world's farmers to acquire more farmland. One might say that all the arable land is used up but more arable land can be created - cut down forests and jungles.
If that's not enough and demand cannot be met, then the market ensures supply goes to those who need it the most - those who don't really want the crops will drop out the market when the price is too high. If the Third World don't buy crops, they clearly don't want to eat as much as the rich want to drive cars.
If the poor are really that attached to eating, they'll find some means of raising the cash required. Migrating to rich countries, doing the jobs the natives won't do and sending money home is a common method. Obviously, any impediments to worker migration is an impediment to the free market and should be removed.
If migrant workers fail to reach countries with worker demand due to immigration laws, then they clearly don't need the money that much. After all, they can always use the black market. It's the basis of the Russian and various East European economies. Indeed, they've already set up people smuggling operations.
In short, rising crop prices are just part of the system. If the poor are starving they're simply being lazy.