To make the filling, gently toss the berries, sugar, lemon juice, cinnamon, nutmeg, ginger, tapioca and a pinch of salt together in a large bowl. Mix very gently to avoid bruising. Spoon the filling into a 9-inch deep-dish pie pan.
To make the crumb topping, stir together the flour, oats, dark brown sugar, cinnamon and salt in a medium bowl. Blend in butter with your fingertips, fork, or pastry blender until mixture forms pea-size clumps.
Spoon topping evenly over fruit filling and bake on a middle rack in a preheated 375 degree F oven until fruit is bubbling and the top is nicely browned, about 60 minutes. Let the crumble cool for about 10 minutes before serving.