After the Cuban Revolution, the Granados attacked the Roth syndicate, starting a war against them in order to seize power in Miami, to make up for Rico Granados' losses in the revolution. Hyman Roth sent one of his men to go undercover with the Granados crime family, but they figured him out and made him a hostage, rendering Roth unable to fight back. Fortunately, Roth heard about the Trapani - Rosatos War and how Dominic had eliminated Carmine Rosato's family, so he called him to Miami to ask for his assistance.
Upon his arrival in Miami, Dominic and his crew saved Steven Ciccoricco, Roth's hostage, and declared war on the Granados family. Roth subtly retreated from the action after this, leaving Dominic to take on Granados alone.
The Granados were fortunately also in a middle of a war with the remaining Rosato Brothers, Tony, which meant they had to fight on two fronts: against Tony and against Dominic. The Corleones immediately attacked and took over several Granados gun smuggling businesses, making them much weaker without strong firearms and lacking strength with better weapons and ammo and their diamond smuggling businesses costing them a great deal of money and stripping them of their supply of bulletproof vests.
After taking a few other businesses, Dominic sought and murdered a few of Granados' made men, so they weren't able to fight back as before.
The final days
Again mercifully for Granados, the Trapanis and the Manganos had a dispute, and Dominic took the only warehouse Mangano had in Florida, Global Storage. They fought back immediately and took over all of the Trapani gun smuggling warehouses and other businesses in the city and the Mangano crime family joined the Miami conflict between Dominic, Tony Rosato and Granados.
When Michael Corleone's trial ended, Dominic and his family focused themselves on the war in Florida and Cuba, and after massive amounts of bloodshed, took over all remaining businesses in Miami and assaulted the Granados Compound, eliminating Rico Granados and seizing it for their own.
The Trapanis eliminated the Granados and took all of their businesses - as well as all other the rival families in Miami conflict. With gained connections, places of influence and money made from controlled shops and warehouses, becoming one of most powerful families on the east coast.
Behind the scenes
The Granados war is event in the video game, so as well as most of families stated here, even Granados crime family do not appear in movie or novel.