I'm aware of no economic theory that claims this, but it is case that countries and people do not grow, but rather certain areas and communities that shift into a capitalist mindset. Most human societies can be characterized by sharp stratifications. One group of course handles agriculture, others fight over wealth, and usually you have some group that does the trade. These are the Jews of Europe, Chinese Hakka in Southeast Asia, Lebanese and Hausa in West Africa, Portuguese in Spanish America, Indian Gujaratis and Punjabis across the Indian Ocean, along with Parsis, Armenians, and many other groups. The pattern for many of these groups is similar--you maintain an outsider status in another area while trading with them, combining knowledge of trade routes with local connections. The advantage is that you get the trust and finance of a close-knit community essential for controlling risk and the potential for theft. While a rich farmer is expected to redistribute his income to family and friends, a rich trader can loan that money out and hire his relations.
This model of trade relies on a starkly pre-modern set of constraints due to the costs of abandoning openness and accessibility. It really only makes sense for economies characterized by a complete lack of trust, third-party interactions, and liquidity, because really trade is one of the simplest forms of arbitrage and should happen frequently. So it's interesting to see the success of this model through times of free trade and to the present day essentially due this system's ability to transmit certain values.
It's popular to complain about British colonialism of Asia as an outright plunder along Burke's lines, and there is much to that. However as far as empires go, it was a relatively light one that encouraged free and open trade--certainly in contrast to intrusions after de Gama leading to the control of maritime chokepoints and a sharp reduction in inter-Asian trade conducted by Asians. The evidence for British openness is given by the success of not only State-approved traders but third party merchants. Be they Baghdadi Jews or Parsis or wandering Scotsmen, there was a lot of money to be made in this part of the world. Beyond trade, you had a big rush towards offshoring manufacturing into these areas as well, including infrastructure and cotton mills. The big question is why India and China didn't gain more from this growth, and why more communities didn't take part. Odeg Galor's answer to the first is that the gains from trade led to greater income per capita in richer countries and greater aggregate population in poorer countries. As far as the second goes, it's tough to say. A regime of relatively open trade and commerce enriched some groups while others failed to take part. Arguably, the inequality of economic power accruing to certain groups is inevitable in a free market and leads to violence when paired with democratic politics.
Okay, both pre-modern economies and colonialist systems impose many constraints encouraging group identities, but the real suprise is that modern economies, if anything, amplify the advantages held by certain groups, largely because of the values these groups possess. This is an amazing time to be alive, with a roughly meritocratic system and outsize opportunities for driven and educated people. So it's amazing to find that people still fail to respond to these incentives. On the other side of the distribution, former advisor to President Clinton William Galston says that you need to do three things to not be poor: finish high school, marry before having a child, and wait until age 20 to have a child. 8% of people who do these things are poor, while 79% of people who do not are poor. Getting into family breakdowns gets into a world of unemployment, spatial mismatch of jobs, and deviant family structures. Men really have no excuse, responding to the responsibilities of being primary bread-winner by falling behind academically and extending adolescence. The media doesn't really provide any good role models here either.
But the broader point is that there are people in a culture of success and other people who are not. These build on long-existing differences in attitudes towards work and education held by different groups in society, and extend on an array of political, cultural, and arguably genetic grounds. While economists like to assume that people are in some sense similar and respond to incentives equally, there is clearly some sort of culturally conditioned response to opportunity. In many ways, the capitalist mindset is inhuman and difficult to adjust to as a whole group. One consequence is that the demographic composition of a standard American elite college in no way mirrors that of the country as a whole, but these people comprise an elite class, and the admissions officers staffing the gates prefer to diversify on the basis of superfluous characteristics while blaming poor schools for colleges' inability to draw from large segments of the population. The great thing about such a meritocratic system is that you co-opt exactly all of the people who would otherwise complain. There has probably been too much hand-wringing about whether there is a "genetic" or "cultural" basis. Cultural connections established at an early age seem rather as firm as those established by genes.
I realize that this is not very original, but I don't see many other people talking about this. This trend doesn't look like it's going away.