Puree the blueberries and strawberries in batches in a blender or food processor. Strain into a medium bowl, through a sieve lined with cheesecloth.
Stir brown sugar into berries. Stir in soy milk until well blended. Pour mixture into small paper cups, filling about two-thirds full. Freeze until beginning to firm. Insert wooden sticks and freeze until firm, 2 to 3 hours.