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|dependencies=mediawiki.util|targets=desktop,mobile|type=general]|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.