To export the database-util gadget, click on "Download" button, save the downloaded file, go to Special:Import on the destination wiki and upload it. Then add the following to MediaWiki:Gadgets-definition page:
* database-util[ResourceLoader|hidden|dependencies=ext.gadget.psw-util,oojs-ui-core,oojs-ui-widgets,mediawiki.widgets.DateInputWidget,jquery.tablesorter]|database-util.js
You must have appropriate permissions on the destination wiki (including the right to edit system messages) and import from file uploads must be enabled.