To export the Carousel gadget, click on "Download" button, save the downloaded file, go to Special:Import on destination wiki and upload it. Then add the following to MediaWiki:Gadgets-definition page:

* Carousel[ResourceLoader|default|type=general|dependencies=mediawiki.util]|Carousel.js|Carousel.css

You must have appropriate permissions on destination wiki (including the right to edit system messages) and import from file uploads must be enabled.