The Great Depression was absolutely transformative for Latin American countries on several levels, though its effects do vary from country to country. In general, its main effect (as BarCasaGringo has pointed out) was the collapse of both the political and economic order that had prevailed since the late 19th century, typically described as oligarchical liberalism centred around export-led economic growth. (In broad strokes: Crecimiento hacia afuerareplaced by crecimiento hacia adentro.) The order that replaced it was (in general) less liberal/more authoritarian, less oligarchical/more populist, more interventionist, far less international/more nationalist, and reflected both the increasing power of the state, and the greater weight of the urban lower classes in political economy.

There were two major shocks in the Great Depression. The first was the near-total shutdown of global capital flows following the Wall Street crash of 1929. American finance had become (since taking over for Britain the end of the first world war) the major source of new investment funds, which had been used by Latin American states to finance rapid infrastructural development (notably but not exclusively railroads) and to "solve" the problem of perpetual government deficits without having to increase unpopular taxes. All Latin American states had a fiscal structure based around taxing imports, which was a politically convenient method of raising funds, but which required high levels of trade in order to maintain state solvency. (Note: this is unlike "deeper" methods of income or consumption taxation, where governments can raise or lower taxes on the domestic population, regardless of the trade situation.)

So, this was the game as it was played in the export-oriented period: secure high levels of investment with the promise of future export earnings; invest that money in high-growth, export-oriented sectors; export primary products to earn foreign currency revenues; tax the resulting imports of manufactured goods to pass the fiscal burden onto consumers and close the fiscal loop. So long as there was stable international demand for primary products, and the rapid pace of economic growth was enough to compensate for the ever-increasing debt burden, Latin America could grow quickly without having to confront painful social, political and economic reforms. (Just how painful and uncertain that process could be, Mexico learned the hard way in the 1910s...)

But all this was contingent on the industrialized world's ever-increasing appetite for primary goods. This model had already started to creak at bit after 1900 through the 1920s, as protectionism started to return to fashion, and as industrial alternatives to primary commodities started to come into existence (chemical dyestuffs, synthetic rubber, chemical fertilisers, synthetic fibres, etc...) which undermined any export economy unlucky enough to specialise in those goods (Manaus, Nitrates and Guano in Chile and Peru, for example). Increased industrial production of foodstuffs was also a double-edged sword, which allowed agricultural regions to increase their production of wheat, meat and sheep, but also allowed their competitors (Aus, NZ, Canada, Russia, US, and European farmers behind tariff walls) to compete with exports.

All this meant that Latin America was already tending towards industrialisation, though in a "staples thesis" sort of way, driven by export processing and urban demand for consumer goods rather than heavy industrial goods. New kinds of politics were arising in this context, sometimes, this was at odds with the oligarchy of the "oligarchical" liberalism of the previous period (One might look at Yrigoyen in Argentina, or the abortive reformism of Madero in Mexico) and sometimes this was simply the breakdown of oligarchical alliances (the Guerra de los Mil Dias in Colombia). But as the world economy came back together in the 1920s, the old export-driven model was at least still vaguely feasible, and so it trundled on without complete collapse (except of course in Mexico, where everything really did fall apart, though also transformed more radically under Lazaro Cardenas in the 1930s.)On the eve of the Great Depression, then, Latin American states were still heavily dependent on international trade and capital flows, not only to sustain the economy, but also the polity.

Then, quite suddenly, it all stopped. The threat of the Hawley-Smoot tariff act started an international trade war. Germany defaulted on its debts, and the resulting standstill agreement froze international credit markets. Countries began leaving the international gold standard, even though international debts were supposedly denominated in gold. Demand for exports dropped to very low levels, causing a simultaneous price and quantity shock. Foreign currency revenues, traditionally obtained by taxing imported manufactures, dropped to very low levels, leaving Latin American governments with almost no choice but to default on their international debts. (The usual consequence, being locked out of capital markets, was far less relevant when those same capital markets were not lending.)

The old system simply did not function in this environment, as the entire fiscal and political basis for the existing regimes evaporated. The incumbent regimes, mostly semi-democratic hybrids of the old oligarchy and the new urban populism, disintegrated or were overthrown. In their place (generally) came populist authoritarians: for instance, Getulio Vargas in Brazil or Lazaro Cardenas in Mexico. Uriburu in Argentina was more authoritarian than populist, but then Argentina was further along in the incorporation of the urban working classes into the political system; Peron would later be a clearer model for Argentine populist authoritarianism. Gaitan never actually comes to power in Colombia, but his anti-oligarchical, populist strain of liberalism becomes a major force in this period, and his eventual assassination set off decades of violence (bad enough to be known as THE violence, la violencia.) In any case, incumbent regimes were discredited, and the replacements were those who could put together a working political coalition under the new circumstances, usually with broad application of charisma and force. (This was hardly unique to Latin America, of course; anyone looking at Europe in the 1930s would not struggle to find "populist authoritarians.")

Economically, Latin American states turned inwards. At first, this was mostly because they had no choice in the matter; nobody was buying what they were selling, and there was no money to borrow, so the price of manufactured imports shot upwards. This collapsed standards of living, at least in the short run, but it also incentivised domestic industrialisation. The relative investment value of an export-oriented farm or mine vs. a domestically-oriented factory changed dramatically in favour of import-substituting manufactures. While the great depression was not the beginning of Latin American industrialisation (that started in the 1890s, really) it was a very powerful shove down that path.

The rise of manufacturing also empowered the urban working classes, whose output was suddenly much more critical than it had been previously, leading to their increasing incorporation in the coalitions of the new governments, and putting the "populist" in populist authoritarianism.

Gradually, these measures became less a matter of necessity and more a matter of policy. Raul Prebisch would bring this vision to CEPAL in the 1950s, along with his famous thesis that the relative price of primary goods would always fall in the long term. (Understandable, if one starts from the 1890-1900 period, and measures until WWII - the trend would have been obvious, though it is less clear that this is true in general.) Latin American states generally moved decisively towards "inward facing growth" and stayed there as a matter of policy preference for decades afterwards, until the late 1960s.

So, the Great Depression had a tremendous effect on Latin America, where it is commonly known as the Crisis of 1929. However, the effect that it had on the region was completely different from the effect that it had on the United States and Europe. To understand why, it is important to understand the attributes of the ruling oligarchic elite of many Latin American countries on the eve of the Great Depression. Starting in the mid-nineteenth century, many countries in Latin America came under the political domination of their own oligarchic elite. These oligarchies were mainly made up of an alliance of large landholders, the owners of import–export houses, and a wide array of other social actors which came to dominate both the political scene and the economies of these countries. These oligarchies were also characterized by their near religious belief in classical liberalism and nineteenth-century positivism, which allowed them to consolidate and justify their support for the socioeconomic order from which they benefited. This particular order was part of the Anglo-centric world market, where the law of comparative advantage dictated that countries like those in Latin America were better suited for the production and export of raw materials, while industrialized nations like Britain had the advantage of being able to convert those raw materials into finished consumer goods. That economic order was then combined with the classical liberal and positivist belief in that very order being seen as not only the best mode of conduct for Latin American economies, but the closest thing to the natural order. The laws that governed this social and economic order upon which the oligarchies rested were believed to be scientifically proven facts of nature, and therefore any attempt to alter the order in any way was seen as illegitimate interference. That’s normally lead to various repressive practices being institutionalized in these oligarchic regimes as a response to social movements like organize labor and calls for political reform. These regimes, including the government of Porfirio Diaz in Mexico between 1876 in 1911, the rule of Julio Argentino Roca’s National Autonomist Party in Argentina between 1880 and 1916, and the colloquially named practice of “política café com leite” in Brazil between 1889 and 1930, we’re all emblematic of this phenomenon.

It was because of these social, political, and economic realities that led to the radically different responses to the Great Depression in Latin American countries than in the United States and Europe. Using the United States as an example, it took several years until classical liberalism as championed by Herbert Hoover was finally discredited, with the electoral landslide achieved by Franklin Roosevelt as indicative of the American electorate’s complete lack of faith in the old economic order and its faith in a laissez-faire response to the Depression. However, this would not occur until a little over three years after the first days of the Depression. Meanwhile, many Latin American countries experienced such change far earlier. The effects of the Depression were such that the old social and economic order we’re almost immediately discredited. If I looked at the political history of Latin America in 1930, nearly every major country experienced some form of political upheaval as a result of the global economic crisis. Argentina, Chile, Brazil, Peru, and Cuba all experienced some form of regime change in one form or another during that single year. Venezuela would not experience such upheaval until a year later. Because the old oligarchic order had prevented their nations’ from industrializing in accordance with the law of comparative advantage, the orders’ downfall brought about efforts at state-directed industrialization and other forms of intervention in the economy, which allowed these states to recover from the effects of the depression at a much greater speed than in the United States and Europe.