As nations become more developed birth rate decreases.
The world is not over-populated, but certain parts of it are. The solution? Do nothing. Overpopulation will naturally sort itself out.
We, however, encourage over-population in those areas by, for example, donating money for food to Africa, the most over-populated continent in the world (it is large, but most of the continent is not arable land). Now, there is easily enough food to go around but we'd have to jump through hoops to get it from the farms producing excess in the Americas to Africa, but we'd be getting nothing in return. Why support someone when they are a burden? It's not natural.
Naturally, famine kills some of them, disease kills some more, war over limited supplies kills even more, and some immigrate to the under-populated countries. Now here comes the problem. Mr. Bigshot overseas rich country has some special interest groups that, for whatever reason (be it religion, humanitarianism, etc), decide they need to save all these people. They show everyone in the country graphic pictures of starving children, create sad commercials, etc. The people of that country decide to pay for the overpopulated country's food, medicine, and the rich country's military might force 'peace' upon the country. That country will then keep accepting immigrants from that country as refugees, eventually naturalizing them. 3 out of 4 natural solutions to overpopulation are no longer functional. But one is, and that is immigration. You have a country reproducing faster than they can immigrate out. The immigrants will keep coming, eventually outnumbering the original inhabitants of Mr. Bigshot country. Culture will be destroyed, racism will become rampant, socially it will be a disaster, the immigrants will be uneducated and may not even speak the language, first-generation immigrants will still maintain a third-world birthrate, etc. Eventually the under-populated country becomes over-populated. They will struggle to support themselves, and will be unable to support the poor overseas country. Then the nonfunctioning natural systems of managing overpopulation will re-activate, but they will do so both in the poor country AND in the rich country, and then the cycle will repeat itself, as another rich bigshot underpopulated country will step up to the plate and begin accepting immigrants while keeping the inhabitants of the original country alive (this whole process can occur in multiple countries simultaneously).
And that is how a few over-populated countries can destroy over-populated countries. The solution to breaking this cycle is to either develop the country to the point where it can support it's population (right now we do little developing, more maintaining), or to just let nature run it's course in the first place.
wut