Bring mint, ginger, sugar, and 2 cups water to a simmer in a small saucepan over medium heat, whisking to dissolve sugar. Remove from heat and let sit 1 hour. Strain through a fine-mesh sieve into a large measuring glass or a small bowl; let ginger-mint syrup cool completely. Cover and chill until cold, at least 30 minutes.
Combine vodka, lemon juice, ginger-mint syrup, and 4 cups water in a punch bowl and stir well.
Ladle spiked lemonade into ice-filled glasses and garnish with lemon slices and mint leaves.