In a large bowl combine blueberries, raspberries, blackberries, granulated sugar, tapioca, cinnamon, ginger, and salt. Let berry mixture sit for 15 minutes, stirring occasionally. In a 2-quart disposable foil pan place the fruit mixture. Cover pan tightly with foil.
For a charcoal grill, arrange medium-hot coals around edge of grill. Test for medium heat over center of grill. Place foil pan on rack in the center of grill. Cover; grill for 30 minutes. Meanwhile, for the topping, combine crushed graham crackers, flour, and brown sugar. Using a pastry blender, cut in butter until mixture resembles coarse crumbs.
Uncover pan. Sprinkle topping evenly over fruit mixture in foil pan. Cover grill; cook for 10 to 15 minutes more or until mixture is bubbly and fruit has thickened. (For a gas grill, preheat grill. Reduce heat to medium. Adjust for indirect cooking. Place foil pan on grill rack over burner that is off. Grill as above.) Cool on wire rack for 20 minutes. Serve warm.
If desired, serve with Beary Good Cheesecake Ice Cream. Makes 6 servings.