Add site branding to external site
I think creating a service REST API to be consumed by the client sites is the cleanest solution. 

This has been implemented in Drupal 7 with sharedblocks but there no Drupal 8 port at this time.  

Studying this module would give you a solid foundation on how to accomplish your task in Drupal 8.

Based on the same concept there is also distributed_blocks module but again does not seem to have a Drupal 8 port.