The economic conflict between free labor and slave labor, of northern capital versus southern agriculture, was the engine of U.S. history from independence up until the ratification of the 13th Amendment in 1865. Did this conflict exist in any other New World slave economies?
This is a great question, and I am afraid that I am stepping a bit out of my area of expertise but can offer some insight since I was educated in Brazil. So this answer only pertains exclusively to Brazil, which is rather a unique case due to its cultural uniqueness within South America.
From what I can gather, Slavery by itself never lead to concerted separatist effort because unlike in the United Staes there was no such separation between states that allowed or disallowed slavery. Brazil had a pretty decently strong centralized government in the figure of the Emperor Dom Pedro I and later his son Dom Pedro II (if you ignore the brief period of instability between Pedro I and Pedro II reign due to the regency) so laws tended to apply evenly across the country. There were no "free states" that enslaved people could run away too, as slavery was rather a national institution and provided the majority of revenue of the country since it depended on Sugarcane and later Coffee for their economy.
I must stress that this does not mean there were no slave revolts or that enslaved people did not resist, much to the opposite. Several of the largest slave revolts/rebellions took place in Brazil. Look at the Quilombo of Palmares in 1605-1694 and the Male Revolt in Bahia in 1835. The vastness and relatively inaccessibility of the Brazilian jungle also facilitated running-away as a mechanic.
Interestingly, the push for the Abolition of Slavery in Brazil did receive actual official support besides the usual Abolitionists (like Jean Baptsite Debret and Joaquim Nabuco). There was huge internal debate over the issue of slavery and the Emperor Dom Pedro II saw Slavery as unattainable and disadvantageous for Brazil which was becoming increasingly isolated due to its stance on Slavery in the late 19th century. So he pushed for the abolishment of slavery throughout the 1870-1880s culminating in the "Lei Aurea" in 1888 officially abolishing the institution of Slavery.
It had the unattended effect of alienating the elites from supporting the monarchy, since most of them were plantation owners relying on slave labor. So a relatively small group of army officers backed by rural and urban elites (for different reasons) overthrew Dom Pedro II in a military coup, despite the majority of the Brazilian population being rather quite fond of the Emperor.
Sources:
Vincent Bakpetu Thompson. Africans of the Diaspora: The Evolution of African Consciousness and Leadership in the Americas (From Slavery to the 1920s). Trenton, NJ: Africa World Press, 2000. pp. 39–44
Glenn Alan Cheney, Quilombo dos Palmares: Brazil's Lost Nation of Fugitive Slaves, Hanover, CT:New London Librarium, 2014
The decline and fall of slavery in Brazil (1850–88) (pp. 113-144) by Leslie Bethell
https://www.jstor.org/stable/j.ctv51309x.8?seq=22#metadata_info_tab_contents
Bethell, Leslie (1970). The Abolition of the Brazilian Slave Trade: Britain, Brazil and the Slave Trade Question, 1807–1869. Cambridge University Press. ISBN 9780521075831.
Schwartz, Stuart B. (1996). Slaves, Peasants, and Rebels: Reconsidering Brazilian Slavery. Urbana: University of Illinois Press. ISBN 0-252-06549-2
Smallman, Shawn C. (2002). "The Overthrow of the Empire". Fear in Memory in the Brazilian Army and Society. University of North Carolina Press. pp. 16–18. ISBN 978-0-8078-5359-7