In a cocktail shaker, muddle the blackberries and basil in the vodka, making sure to get the blackberries to fully release their juices. Add the lime juice to the shaker and fill three quarters of the way with ice. Shake until chilled, about 15 seconds. Fill copper mugs with lightly crushed ice. Double strain the mixture into the mugs through a fine mesh strainer to ensure that no seeds wind up in the drinks. Top with ginger beer and garnish with additional blackberries, basil, and lime slices.