In the 1960s the Total Fertility Rate of the poor countries, the Third World in the terms of the day, was about 6 children per woman. Today, poor country fertility stands at 2.9. China has led the way with an aggressive one-child policy, which has pushed their TFR down to 1.8. India, with less coercion, has gone from about 5 children per woman down to 3, and falling. These are the world population leaders, the "population billionaires" as Ben Wattenberg calls them in Fewer. China and India have worked hardest to get control of their population growth as a matter of urgent national policy.
Yet many other developing nations have also seen their fertility drop dramatically. Brazil and Mexico have TFRs that are probably below the replacement level of 2.1. The rest of Latin America is not far behind. Several of the "-stans" of the former Soviet Union are down to replacement level. Black Africa is the standout in world fertility rates, but even there the TFRs are below their '60s peak. Moreover, because of AIDS and other endemic diseases, African mortality is actually increasing - the Malthusian dark side of population control.
Controlling poor country fertility is the single most important lever in controlling world population. Now that the rich countries are starting to grasp that their populations are likely to start falling, they look to the Third World for their future workers and parents. Yet if poor country populations drop, they will be sending fewer immigrants to the rich countries. Moreover, most poor countries are getting richer, especially the ones with more honest governments. This means that poor people who are willing to move to make a better life are more likely to move within their own country, rather than push on to a job-rich, population-poor country.