Make the rosemary-infused rum: Place a sprig of rosemary in each of two 16-ounce mason jars. Pour 2 cups rum into each jar and close tightly. Let infuse for 2 to 3 days at room temperature.
Make the ginger syrup: In a small pot over medium heat, combine the sugar, ginger and 1 cup water. Bring to a boil, then reduce the heat and simmer, stirring, until the sugar is dissolved, about 5 minutes. Let cool 20 minutes, then strain. (You can also skip this step and use store-bought ginger syrup.)
Mix each drink individually: Rub the rim of a copper cup with maple syrup, then dip in turbinado sugar. Fill a cocktail shaker with 1/4 cup ice. Add 1/4 cup rosemary-infused rum, 1/4 cup ginger syrup and 1/3 cup punch or soda. Stir, then add ice to the copper cup and pour in the drink. Squeeze a few lime slices into the drink, then garnish with a rosemary sprig, cherries and a slice of lime.