Spirit theme, multisite - ability to create a master template

I'm creating a set of 6 websites for my business. They will each have different, but complementary content, and will be connected by links. I'd like them to all have a very similar style/branding. It would be so convenient if I could create a master template that I could simply copy and fine-tune for each site. Additionally, it would be great if changes to the master template would automatically load to each site to save on updating time.