“Throwin’ Down” is an old-time expression for drawing a handgun—typically a revolver—and demonstrably aiming the piece at a live target. I think it may have originated in the south and migrated west as the frontiers were conquered. It conveys an impression of a kind of flamboyance, but also one of deadly intent, as when… “Earp threw down on Curly Bill.” Sure, the term is slang, but slang is what makes our English language so expressively colorful. There are numerous examples of gun terms becoming slang terms with much broader meanings. How about “Don’t go off half cocked” to indicate a lack of preparation or “Lock, stock and barrel” to suggest completeness.
There may be more than to “Throwin’ Down” than we realize. In the early days of caplock revolvers, percussion caps on the chamber nipples were a necessity. They were manufactured in huge quantities and sometimes quality control was lacking. The caps would often shatter and drop particles into the various spaces at the closed rear end of the cylinder and then into the action. In an effort to get this debris out of the gun, some shooters would pull the revolver clear back by their head and then cock it with a down and forward throwing motion. When the revolver was upside down, above the shoulder, the debris from previous shots could drop free of the gun. The term persisted, even if the caplock revolver did not.