I noticed a bug today when using the customizer with domain mapping. It seems that when the settings for "Administration mapping" is set to "Original domain" that the user can't change pages in the customizer (i.e. if they click on the link to another page, it doesn't work, OR if they try to access the customizer from a page other than the front page for their site, then it opens up on the front page).
I am running a sub-domain install.
Just thought I'd flag it up,