A proper answer to this would be book length as there are many different countries in Latin America and not all countries followed the same historical trajectory during this time. To look at what many countries had in common, I will briefly discuss Mexican and Argentine history during this time.
The basic trajectory of these countries' history was that they went through long periods of internal conflict that were finally resolved. They then went on to form somewhat autocratic but modernizing governments. Finally, in the 20th century, they did move somewhat more towards democracy.
In both cases, the internal unrest was based largely on conflicts between federalists and centralists. One side wanted power to be split up among the different areas of the country while the other wanted a strong central government. In both cases, what came after was a fairly autocratic system in which the government provided relative stability and then brought in foreign help to try to develop a modernized economy. Finally, these countries moved towards more of democracy, as seen, for example, in Mexico's revolution in the 1910s.
This is a major generalization, though, and does not clearly apply to all Latin American countries.