Put the dried cranberries in a small bowl and add enough simple syrup to cover. Let soak until slightly softened, about 1 hour. Meanwhile, chill 4 copper mule mugs in the freezer.
For each drink, combine 1 1/2 ounces vodka, 1 ounce each orange juice and cranberry juice and 1/2 ounce simple syrup in a cocktail shaker filled with large ice cubes and shake vigorously. Add 1 ounce ginger beer, seal the shaker again and gently twist the shaker back and forth once (this will combine the drink without making the ginger beer go flat). Strain into a chilled copper mule mug and fill with fresh ice. Garnish with a few of the soaked cranberries.
To make simple syrup, heat equal parts sugar and water in a saucepan until the sugar dissolves; let cool. Store in the fridge for up to a month.