In separate batches, heat [specific fruit], mint, and sugar on medium-high heat until the fruit is soft. Do not add water.  Mash the fruit with a spoon or, in the case of pineapple, blend in a Cuisinart until smooth.  Softer fruit does not need to be pureed.

Puree Method

In a cocktail shaker, squeeze lime over mint leaves, muddle with a wooden spoon, add sugar, fruit puree, and rum, and blend to combine. Fill the shaker with ice, place the lid firmly on top, and shake the drink several times. Pour the mixture into the cocktail glass, leaving headspace, and fill to the rim with club soda. Garnish with fruit, lime wedges, and mint leaves.