Begin by preheating the oven to a temperature of 350°F (175°C). Concurrently, prepare a 13 x 9-inch baking dish by lightly greasing it with either butter or a non-stick cooking spray.

Remove the crescent rolls from their packaging and carefully lay them flat, ensuring they are separated into the pre-cut triangle portions.

On the wider section of each triangle, place a measured amount of apple pie filling. Proceed to roll the dough, starting from the filled end and sealing the apple filling securely inside.

Position each filled crescent roll within the previously greased baking dish. Ensure even spacing and continue this procedure until all of the dough triangles have been filled and placed.

In a separate mixing bowl, blend the unsalted butter and brown sugar. Use a mixer or hand whisk until the two ingredients amalgamate into a smooth consistency, approximately taking 2-3 minutes.

On each crescent roll, place a measured teaspoon of the blended butter and sugar mixture. Following this, pour the lemon-lime soda into the baking dish, ensuring it fills the gaps between the rolls. It is imperative not to pour the soda directly over the crescent rolls.

Transfer the baking dish to the preheated oven. Allow the apple dumplings to bake for a duration of 35-40 minutes or until they achieve a golden brown hue. Upon removal from the oven, it is recommended to serve the dumplings warm. For an enhanced taste, they can be paired with whipped cream or vanilla ice cream.