Grab a large pitcher or mason jar that has a lid and can fit into a freezer (or two smaller ones). Add the raisins, rum, raisins, and cinnamon sticks to the pitcher, and allow it to sit for 1 hour.
To a blender, add the sweetened condensed milk, cream of coconut, coconut milk, evaporated milk, vanilla, nutmeg, and ground cinnamon. Puree until smooth.
Pour the pureed mixture into the pitcher(s), and chill in the fridge for at least 4 hours more.
Rim the serving glasses with cinnamon sugar. Shake the coquito and pour into the glasses. (You can strain the raisins and cinnamon sticks or add some of the raisins into each glass.)