So ...

First you need to export your current customizer settings from the "old" site.

Go to Appearance => Customize and click the "Export/Import" tab to access the available settings and options:

And usually there are no additional settings here. All you have to do is to click the "Export" button:

The plugin will now export all your customizer settings and send them to your browser in a .dat file. If you'll see a small prompt window (depending on your browser settings) just choose "Save File" (instead of "Open File") ...

And just to clear the air ... The plugin will export any theme-related settings stored in the current Wordpress database (color settings, layout directions, header media, etc), but won't export things like menus, site title, widgets, etc.

OK. Let's move forward to the last lesson to see the import process ...