The Customizer Export/Import plugin allows you to export or import your WordPress customizer settings from directly within the customizer interface!
Exporting customizer settings is easy. Click the Export button from within the Customizer and a file will automatically begin downloading with your settings. Export files are named after your theme and can only be used to import settings for the theme or child theme that they came from.
Importing customizer settings is just as easy. Choose the export file you would like to import, select whether you would like to download and import images (similar to importing posts), and finally, click the import button. Once your settings have been imported the page will refresh and your new design will be displayed.
The plugin details: