Sprinkle gelatin over 1/2 cup cold water in a small bowl and let stand until softened, about 5 minutes.
Meanwhile, in a food processor, purée 1 1/4 cups boysenberries or 3/4 cup each raspberries and blackberries, 1/4 cup sugar, and the lemon juice. Strain purée into a small saucepan, pressing to extract liquid, and discard solids in strainer.
Stir gelatin mixture, then spoon 2 tsp. into strained purée (set remaining gelatin mixture aside). Cook berry mixture over medium heat, stirring often, until steaming but not simmering, about 3 minutes. Set sauce in a bowl of ice with a little water until cold. Remove from ice bath.
Heat remaining 1/3 cup sugar and the milk in a medium saucepan over medium-low heat, whisking frequently, until sugar dissolves, 2 to 3 minutes. Add remaining gelatin mixture and cook, whisking often, until steaming, about 2 minutes. Set in ice bath, whisking often, until thickened but still pourable, 9 to 10 minutes; watch carefully, as it thickens quickly.
Meanwhile, beat cream and vanilla in a large bowl with a mixer until thick and silky. Fold in thickened milk mixture.
Pour about 1/3 cup mousse into each of 6 small glasses or canning jars. Drizzle each with 1 tbsp. berry sauce. Repeat layers, using all of sauce. Spoon any remaining mousse on top. Draw a table knife through layers to swirl. Cover and chill until set, about 3 hours.
Sprinkle desserts with crumbled shortbread and remaining berries.
Make ahead: Through step 6 up to 1 day, chilled.