Preheat oven to 350° with a rack in lowest position. Working in 2 batches, pulse apples in a food processor into pea-size pieces. Transfer to a large mixing bowl. Put cherries, cranberries, and raisins in food processor and whirl just until fruit starts to break up. Combine with apples.
Stir in currants, spices, salt, zests, sugars, cornstarch, and rum. Pour mixture into prepared pie pan and spread level.
Lay 1 dough strip horizontally on top edge of pie. Then lay 1 strip vertically on left edge of pie, with the strips meeting at the corner. Lay strip 3 of dough 1/2 in. below and parallel to first strip. Then lay strip 4 of dough 1/2 in. to the right and parallel to strip Continue, working from top to bottom and left to right, until lattice is complete. Trim excess dough flush with pan edge, then use floured hands to press edge of pie and seal strips to bottom edge.
Bake pie until golden brown and bubbling, 1 1/2 hours (you may need to put a drip pan under pie for last 30 minutes). Let cool to room temperature to serve.
Make ahead: Prepare filling and chill up to 1 day. Bring to room temperature while rolling out dough, then bake as directed.