The book has sections about Mexico as well. The basic gist is that the fundamental institutions that result in the inequality have persisted, even if the government has been overturned several times. The overarching theme of the book is about what types of incentives result from various types of institutions, and how positive and negative feedback loops can form therein.
Ultimately, Mexico has suffered significantly from the government granting monopoly rights to particular businesses (friends of the government) and by not respecting property rights. Both of these trends have spanned governments and go all the way back to the Spanish colonialists.
There is little incentive in Mexico to innovate and invest, because if you are too successful, you will likely run afoul of some wealthy friend of a politician who will use his power to ruin you. From the point of view of the political elite, this makes sense, because they are all rich and in charge. But this behavior impoverishes society on average.
Concrete examples in the book include the banking industry, which was effectively monopolized by 2 banks with government support in the 19th century. Contrasted with hundreds of banks with fierce competition in the US in the same time period. This resulted in high interest rates in Mexico (and thus little incentive to invest) and low interest rates in the US (which was rapidly industrializing).
A more modern example is comparing Bill Gates vs. Carlos Slim, who were vying for the title of the world's richest person when the book was written. Bill Gates made his fortune by founding an incredibly successful company. When it was engaging in monopolistic behavior, it was successfully prosecuted by the government. Contrast this with Carlos Slim who made his fortune largely by being granted favorable monopolistic deals by the Mexican government.