I don't know the history of the rest of Latin America independences but I can tell you the Brazilian history. English isn't my first language so sorry for any mistakes.
After the Napoleon troops invaded Portugal in november 1807, heading to Lisbon, the prince Dom João decided to transfer the court to Brazil. Upon arriving he issued several measures that effectively ended Brazil colonial status. Dom João VI in the following years tried to better integrate Brazil and Portugal and, in 1815, promoted Brazil to "United Kingdom to Portugal and Algarves".
In 1820, however, Portugal had a liberal revolution. The Portuguese pressed for the return of the king to Portugal, in part due to their own internal crisis. The portuguese desired to return Brazil to the status of colony. In the end of 1820 a Court created for writing a Constitution with representatives of all of the Portuguese world In Rio de Janeiro Portuguese defended the king return to Portugal and There were Brazilians and "adapted" Portuguese, those whose interests aligned with the colony, that desired and worked for the stay of the king. Dom João, however, feared losing the crown so he went to Portugal, leaving his son, Pedro, as regent prince.
Once the king returned, there were elections for the representation of Brazilians in the Court. The conflicts between the brazilian and portuguese interests were too big and the independence option got more and more interesting.
Brazilians focused on keeping Dom Pedro in Brazil. His decision to stay was consagrated on the 9 of January of 1822 also known as the "Dia do fico"^1 . The country wasn't yet independent. That came only after the prince, upon the arrival of dispatches revoguing his decrees, gave the famous "grito do Ipiranga"^2 saying "independencia ou Morte!"^3 7 of september of the same year.
That's how and why Brasil became/remained an empire after it's independence. Source:
Fausto B. História concisa do Brazil. 3ed, 2018.
Day of I stay
Yell of Ipiranga
Independence or Death!