In a large pitcher, combine the unsweetened cranberry juice, freshly squeezed orange juice, and apple juice. Stir thoroughly to ensure that all the juices blend seamlessly.
Gently add the fresh mint leaves to the mixed juices. Allow them to steep for a few minutes to infuse their refreshing flavor into the mixture.
In a serving jug, fill it halfway with ice cubes. Carefully pour the prepared juice mixture over the ice until the jug is almost full, leaving a little space at the top.
Add the sparkling water on top of the juice-filled jug for a delightful fizzy finish. Use a long spoon to stir gently, ensuring to combine without losing the bubbles.
Enhance the visual appeal by floating the thin slices of lemon, orange, and apple in the punch. This not only garnishes the drink but also adds an inviting aroma.
Serve chilled in individual glasses, topping each glass with a sprig of mint for a burst of freshness.