You cannot directly transfer DLC from one original Xbox to another. DLC is tied to your Xbox Live account, not the console hardware, and the original Xbox lacks a built-in license-transfer tool.
Original Xbox Live services were discontinued in 2010. The Marketplace is no longer accessible, so re-downloading is effectively impossible today. If you still have the original console, your DLC remains playable there—but it cannot be moved to another original Xbox.
For modern Xbox consoles (360, One, Series X|S), license transfer is supported via the "License Transfer" tool in settings.