Combine the pineapple juice, orange juice, and cranberry juice concentrate in a medium saucepan over low heat.
Add the miniature marshmallows to the juice mixture once it begins to steam, but before it reaches a rolling boil.
Whisk the mixture continuously for 5 to 8 minutes, or until the marshmallows have completely dissolved and the liquid is smooth and slightly thickened.
Remove the saucepan from the heat and allow the base to cool to room temperature, approximately 20 minutes.
Transfer the cooled mixture into a large pitcher and refrigerate for at least 1 hour to ensure it is thoroughly chilled.
Just before serving, stir in the chilled sparkling water to add effervescence.
Fill serving glasses with crushed ice and pour the punch over the top, garnishing with fresh strawberry slices.