Preheat oven to 375 degrees F (190 degrees C).

Combine gingerbread cake mix, water, and eggs in a large bowl. Stir until well blended. Divide batter between 2 ungreased 9-inch cake pans.

Bake in the preheated oven until a toothpick inserted into the center comes out clean, 20 to 25 minutes. Place cake pans on wire racks to cool for 10 minutes. Invert and let cool to room temperature, about 20 minutes.

Combine milk and vanilla pudding mix in a saucepan over medium heat. Bring to a boil, stirring constantly. Remove from heat; let cool until thickened, about 10 minutes. Stir in pumpkin pie filling, brown sugar, and cinnamon.

Crumble 1 gingerbread cake into the bottom of a large punch bowl. Pour 1/2 of the pudding mixture over the gingerbread. Add a layer of whipped topping. Repeat with remaining gingerbread cake, pudding mixture, and whipped topping. Sprinkle gingersnaps on top.

Refrigerate until pudding is set and flavors combine, 8 hours to overnight.