https://wordpress.stackexchange.com/questions/99472/how-to-export-import-theme-customizer-settings
Theme mod settings are stored as a single database entry in the options table.
For example, If I look in my wp_options table in a localhost install, and find the key of theme_mods_twentythirteen, then this is the content of it:
a:3:{s:16:"header_textcolor";s:6:"220e10";s:12:"header_image";s:84:"http://localhost/wptrunk/wp-content/themes/twentythirteen/images/headers/diamond.png";s:17:"header_image_data";a:3:{s:3:"url";s:84:"http://localhost/wptrunk/wp-content/themes/twentythirteen/images/headers/diamond.png";s:13:"thumbnail_url";s:94:"http://localhost/wptrunk/wp-content/themes/twentythirteen/images/headers/diamond-thumbnail.png";s:11:"description";s:7:"Diamond";}}
That’s just saving the header image and colors and such for the twentythirteen theme. Copy that to another install and you copied all the theme mod settings for that theme.
