Wash the apple, orange, and lemon well, then slice each one into thin rounds. Keep a few extra slices aside for garnishing individual glasses later. Thin slices float better and release flavor without overpowering the punch.
Fill a bundt pan or ring mold with water, tuck in a few fruit slices and mint leaves, and freeze for at least four hours or overnight. This step keeps your punch cold without diluting it the way loose ice cubes do.
In your large pitcher, combine the apple juice and cranberry juice and stir well. This base can be made and refrigerated a day ahead, which makes party-day prep so much easier.
Place the juice base, ginger ale, and lemon-lime soda in the fridge for at least an hour before serving. Cold ingredients mean less melting ice and a punch that stays fizzy much longer.
Right before your guests arrive, pour the chilled juice base into your punch bowl. Gently add the ginger ale and lemon-lime soda, stirring just once or twice to combine without losing the fizz.
Slide in your ice ring, or add ice cubes if you skipped that step, then float the apple, orange, and lemon slices on top. Tuck in the cinnamon sticks and mint leaves for a pretty, fragrant finish.
Ladle the punch into glasses right away so everyone gets it while it's cold and bubbly. If the bowl sits out for a while, give it a gentle stir before refilling cups.