We should look first at the period between 1914 and 1930, in particular the economic problems of Latin America, and the growth of political threats to elite rule. How strong were Latin American economies and political institutions when the Depression struck? How successful were the economic measures introduced to deal with the Depression? What were the political consequences of the Depression (why were there so many military coups, what sort of regimes resulted, and to what were extremist and nationalist movements stimulated by the Depression)? I would suggest reading some general work then concentrating on one country (don’t forget to look at textbooks on individual countries too).
C. Marichal, A Century of Debt Crises in Latin America: from Independence to the Great Depression
R. Thorp, 'Economy, 1914-1929', in L. Bethell (ed.), Latin America: economy and society, 1870-1930, pp. 57-82
M.L. Conniff, Latin American Populism in Comparative Perspective, esp. chaps. 1, 2, 4, 6
R. Thorp (ed.), Latin America in the 1930s, esp. chaps 1 and 2 (general), 3 (on Chile), 6 (Argentina and Brazil), 8 (Argentina)
M. Falcoff & R. Dolkart (eds.), Prologue to Peron: Argentina in depression and war, 1930-1943
R. Gravil & T. Rooth, 'A Time of Acute Dependence: Argentina in the 1930s', Jnl. Eur. Econ. Hist. 7 (1978), 337-378
A.L. Potter, 'The Failure of Democracy in Argentina, 1916-1930': an institutional perspective', JLAS 13 (1981), 83-109
P.H. Smith, 'The Breakdown of Democracy in Argentina, 1916-1930', in J. Linz & A. Stepan (eds.), The Breakdown of Democratic Regimes
A. Spektorowski, ‘The Ideological Origins of Right and Left Nationalism in Argentina, 1930-1943’, Jnl. Contemporary History 29 (1994), 155-184
G. di Tella & D.C.M. Platt (eds.), The Political Economy of Argentina, 1880-1946
J.M. de Carvalho, 'Armed Forces and Politics in Brazil, 1930-1945', HAHR 62 (1982), 192-223
S.R. Duncan Barreto & J. Markoff, 'The Limits of the Brazilian Revolution of 1930', Review 9:3 (1986), 413-452
S.E. Hilton, 'Vargas and Brazilian Economic Development, 1930-1945', Jnl. Econ. Hist. 35 (1975), 754-779
R.M. Levine, 'Perspectives on the mid-Vargas Years, 1934-1937', JIASWA 22 (1980), 57-80
R.M. Levine, Father of the Poor? Vargas and his Era
T.E. Skidmore, Politics in Brazil, 1930-1964
J. Wolfe, ‘Father of the Poor or Mother of the Rich? Getúlio Vargas, industrial workers, and constructions of class, gender and populism in São Paulo, 1930-1954’, Radical History Rev. 58 (1994), 80-111.
P.W. Drake, Socialism and Populism in Chile, 1932-1952
D. Corkill, 'The Chilean Socialist Party and the Popular Front, 1933-1941', Jnl. Contemp. Hist. 11 (1976), 262-274
J.A. Etchepare & H.I. Stewart, ‘Nazism in Chile: a particular type of fascism in South America’, Jnl. Contemp. Hist. 30 (1995), 577-605
R. Tardanico, 'The Mexican State and the World Crisis, 1929-1934', Review 9 (1986), 453-482
O. Ciccarelli, ‘Fascism and Politics in Peru during the Benavides Regime, 1933-1939: the Italian perspective’, HAHR 70 (1990), 405-432
S. Stein, Populism in Peru: the emergence of the masses and social control
R. Thorp & G. Bertram, Peru, 1890-1977: growth and policy in an open economy