In a small saucepan over high heat, add 1 cup water and the sugar and heat to a simmer. Stir to melt the sugar, turn off the heat, and allow the simple syrup to cool.
Cut the lemons and limes into 8 pieces.
In the bowl of a food processor fitted with the bottom blade, add the lemons and limes and process until finely ground, about 2 minutes. Add in the simple syrup, pulse a few times, add in 2 cups of water and pulse again. Add in the raspberries and pulse to combine.
Strain through a fine mesh strainer into a glass bowl or pitcher. Add the remaining 2 cups water and chill.
Add 1/2 cup ice and about 10 mint leaves to a cocktail shaker. Muddle the mint, add in 8 ounces of lemon-lime-raspberry mixture, and 2 ounces vodka. Shake to combine. Strain into a glass filled with ice and garnish with mint, a lime wedge, and a couple of fresh raspberries. Repeat as needed.