The site had been migrated from "tmg.alakartsites.com" to "themalinggroup.com". The original site still works fine.
On the migrated site (the new one) initially when working with Customizer, there were some mixed content erorors in the console because some URLS were not updated during migration (migration was done with All in One WP Migration plugin). Velvet Blues Update URLs helped for that.
There's another issue though. Some changes made via customizer are not reflected on site and break other changes. As an exammple: try adding this code:
<i class="fa fa-mobile" style="font-size: 24px; vertical-align: middle;"></i>  <font size="2">858.614.0253
<i class="fa fa-envelope" style="font-size: 19px; vertical-align: middle;"></i>  <font size="2">email@example.com
to the top bar, then save it (you'll see that it's working on preview) and try to make other change - e.g. change font in "General -> Headings" section.
That happens only on that migrated site while original one is still working fine (you can see effect of this code there, see top bar).
Support access granted.