Pour the vodka and lime juice into a 1-gallon drink dispenser.
Stir for about 15 seconds until fully combined.
Cover and refrigerate the dispenser for at least 1 hour and up to 24 hours.
Just before serving, add ice to the dispenser up to the fill line.
Fill each copper mug with ice, then pour the vodka-lime mixture from the spigot until the mug is about half full.
Top each mug with chilled ginger beer, stirring once with a bar spoon to lift the lime to the surface.
Garnish with a lime wheel and a mint sprig and serve right away.
Good to Know
For the zero proof mug, skip the dispenser entirely and build each glass with lime juice, ice, and ginger beer alone. The ratio of lime to ginger beer stays the same.