This really has nothing to do with the theme customizer. From the Codex on moving WordPress :
Existing image/media links uploaded media will refer to the old folder
and must be updated with the new location. You can do this with the
Velvet Blues Update URLs plugin, or with a search and replace tool, or
manually in your SQL database.
The values in the database (where your logo’s URL is stored… nor the URLs for images that are inserted into posts) are not automatically changed. These must be changed manually, or you can use certain plugins/scripts to automate the process.
I have used both the Velvet Blues Plugin and the Serialized Search and Replace script with much success. If I had to pick, I’d say that Velvet Blues is easier to use for people who don’t/can’t do FTP access.