Place the pieces of citrus and the sugar in the bottom of a pitcher. Using a muddler or the handle of a wooden spoon, crush and mash the fruits to release the oils and juice from the rind. When you have some nice juice in there and the sugar begins to dissolve, add the liquor and stir to combine. Fill lowball glasses with ice cubes and pour the cocktail over the ice. Do not remove the pieces of crushed fruit. Garnish with mint and lime. Use fresh sugarcane sticks as stirrers, if desired.