Hello there.
Sorry for the late response.
All tables created with wpDataTables Lite are non-server-side tables, and are linked to an existing data source. Those tables’ structure is saved in wp_wpdatatables, wp_wpdatatables_rows and wp_wpdatatables_columns database tables. In order to move them to a new website, you need to copy these tables, and paste them in the new database, providing the correct link to the external sources on the new website.
If the external source is Google Spreadsheet, XML, PHP and JSON, you won’t have to do anything, but if they are generated by CSV or Excel, you will need to upload the files in ../wp-content/uploads/ of your new website, and update the files’ location in wp_wpdatatables database table.
You can, of course, use the Duplicator plugin, which will do this for you, but it will copy the entire website.
Kind regards.