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The Chiappa Firearms X-Caliber 12ga/9mm Gauge Adapter Insert is designed for use with 9mmx19 ammunition. While, yes, you could shoot a 380 through a 9mm barrel the headspacing is different. 380 is 2mm shorter and will have to make a "jump" those extra 2mm which will cause wear and gas cutting. Overtime it will cause more damage and not allow a standard 9mmx19 to seat properly causing dangerous pressure spikes.
No. Ignore that other answer. The .380 will NOT have to "jump" any distance. The reason you can't use a .380 in a 9mm chamber is that both cartridges headspace off the case mouth. The .380 will sit 2mm deeper into 9mm chamber. Which means that in most cases, the FIRING PIN won't reach the primer to be able to actually ignite the cartridge. IF by chance the firing pin is able to reach the primer and ignite the cartridge, the casing would wind up being thrust backwards 2mm and the bullet itself would then yes, have to "jump" those 2mm to the bore causing errosion of the chamber throat. Overall short answer is no, do not attempt to fire .380 from a 9mm chamber.