The East Asian economies that did well during the post-war period are often described as following the "Japanese model". That is to say, they featured high investment, (relatively) low domestic consumption, export-oriented growth, and political stability--even if political leaders changed, the long-term economic growth model did not.
One thing I believe is often overlooked, however, is the role of the US as an export market. During the early 1950's, some American leaders worried that the loss of what was traditionally one of Japan's greatest export markets, China, to communism would lead Japan (and, to a smaller extent, South Korea and Taiwan) into some sort of accommodation with the Soviet Bloc. That's why the efforts to close off the US market to Japanese exports seen in the 1930's, for example, did not repeat themselves during the 1960's and 1970's, even though you had some US industries (textiles, for one) loudly campaigning for it. American national leaders were willing to pay the price--domestic political unpopularity--to keep the East Asian "tiger" economies on board as allies.
I think, then, to have other countries join the ranks of the "tigers", you need three things. First, the countries have to abandon the then-current economic ideas of import substitution, and move towards an export model. Second, you need these countries to have enough political stability to allow for high levels of long-term investment and long-term economic planning. Third, you need to somehow have the US encourage exports from these countries, through the use of export credits and such. That last one seems the hardest to me. You need to somehow convince both the US and local leadership that the way to fight communism in Latin America, say, is through economic growth and not right-wing autocracy.
Unfortunately, I don't know enough history to offer specific PoDs for the countries in question.
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