Purée the blueberries and strawberries in batches in a blender or food processor. Strain into a medium bowl through a sieve lined with cheesecloth or muslin.
Stir brown sugar into sieved berries. Stir in soya milk until well blended. Pour mixture into small disposable paper cups, filling about 2/3 full. Freeze until beginning to firm. Insert wooden lolly sticks and freeze until firm, 2 to 3 hours.
To remove from cups, tear the paper cup or place under warm running water for a few seconds to loosen.