You can import export your theme settings including customizer panel settings. If you want to switch to a child theme, it is recommended to export your settings first.
Note : In a WordPress Multisite you need to enable importing of the txt file extension the superadmin area settings.
Note : If for some reason the settings in the customizer look different then on the real view of the website then try the following:
1) Export the customizer settings
2) Reset the customizer settings
3) Import the customizer settings.