Is there a way to move an asset from one Organization or Case to another?

Moving an asset from one Organization to another is prohibited.

Organization assignment is performed on the installation of the AIR responder process, and assets are directly bonded to the associated Organization. So there is no way to directly move an asset from one Organization to another. If you want to move an asset from one Organization, you must uninstall the responder from the asset, and then you must create a new responder deployment package and must select the Organization you want.

Moving an asset directly from one Case to another is not a valid action.

Actually, assets are not directly attached to Cases. They attach to a Case via a Task. When the Tasks of the assets are added to the Cases, the assets appear in the Case. An asset can be included in more than one Case. When you add the asset's Tasks to different Cases, the asset appears in those Cases. You can add Tasks to a Case or send a completed Task to other Cases with the "Send to case" action.

If you remove all the Tasks of the asset from the Case, the asset will be removed. You can directly remove the asset from the Case. This way, all its Tasks will be removed from the Case. Then you can add Tasks in interest to any Case you want.