Hello. It's a bit of an older answer, but here is my reply to this on Cuba.

The short version is, there were separatist movements in the Spanish Caribbean, but they were largely stomped out. Mainland independence movements tried to send aid, but international politics, domestic discontent, and the devastated state of their economies prevented those efforts from coming to fruition.

Like u/ThucydidesWasAwesome mentions, there were separatist movements in Puerto Rico and other Caribbean Spanish colonies. These efforts were quickly suppressed by Spanish authorities. The most famous movement in Puerto Rico is known as El Grito de Lares. This book (in Spanish) was written in 1872 by José Pérez Morís, who was opposed to the independence movement in PR, is a good source. A good summary (in English) can be found here.

El Grito de Lares was planned by Ramón Emeterio Betances and Segundo Ruiz Belvis. At the time, Spain was involved in different wars for independence in some of their other colonies. Meanwhile, in PR, pro-independence sentiment was spreading (although is did not have a majority) due in part to crushing taxes and tariffs imposed by Spain (to fund their expenses elsewhere). In 1865, Spain established the Informative Board on Overseas Reforms in order to receive complaints from the colonies. This proved to be unsatisfactory for the PR representatives, who quickly realized that the Board had a majority of mainland Spain-born delegates. The Spanish delegates would vote down almost every measure the colonists suggested, including the measure on the abolition of slavery.

The discontent following the Board acted as the catalyst that spawned the Grito de Lares. Betances had also written The 10 Commandments of Free Men from his exile in St. Thomas in 1867, which was also great inspiration for the rebels.

In early 1868, Betances and Ruiz Belvis, who were then exiled to the Dominican Republic, created the Revolutionary Committee of Puerto Rico. The insurrection was planned by some exiled revolutionaries as well as citizens mainly located in the western mountainous region of PR. On September 20, 1868, members of the Revolutionary Committee met and planned the insurrection, which was to take place on September 29 of the same year. Nevertheless, their plans were discovered by a Spanish captain who then arrested the members of the Camuy cell of the Committee. Meanwhile, Spanish authorities prohibited a ship containing weapons and supplies for the insurrection from leaving the port in the Dominican Republic. Fearing exposure and arrest, the leaders of the insurrection in Puerto Rico decided to change the date of the Grito to September 23. They did not inform Betances.

On September 23, 1868, the insurgents took the town of Lares and proclaimed the Republic of Puerto Rico. The next day, the forces of the insurrection were preparing to take the next town, San Sebastián del Pepino, when they were greeted by a strong militia assembled by the colonial government. The militia rounded up the insurgents and the Grito was now considered over.

Initially the insurgents were imprisoned and some sentenced to death, but in 1869 the Spanish governor issued a general amnesty, although some prominent leaders were exiled.

In the aftermath of el Grito de Lares, other smaller attempts at insurrection occurred around the island, but none were successful. The Revolutionary Committee continued its work from exile. On the island, prominent members of the PR society lobbied Spain for increased rights. In 1897 the liberal Spanish government agreed to grant limited self-government to the island by royal decree in the Autonomic Charter.
This self-government was extremely short-lived, and in 1898, the Spanish-American war leads to the invasion of PR by the US, and eventually, Spain cedes Puerto Rico to the US, thus ending Spanish rule on the island.