To export from Tropy, right-click on an item in the item table. From the right-click menu, select Export Item > JSON-LD. This action will open a dialog box, where you can save your JSON file onto your computer. To export multiple items, Ctrl/Command+click or Shift+click to select multiple items. Then right-click and select Export Selected Items.
At present, Tropy only exports metadata to JSON-LD, not associated photos. Your photos still exist outside Tropy as well, since importing in Tropy copies your photos.
Export to Omeka S occurs through the Omeka plugin. In order to use it, you will need an existing Omeka S installation and access to the API information.