How can multiple sites use a common default theme with site specific customizations?
Consider starting from a Base theme, such as Omega, Zen, Bootstrap, etc. Or build your own (custom) theme that you can use as a Base theme (that would be youru "core theme".

And then create Sub-themes (that would be what you named "their custom themes").

Note: the question about "Right way to start with a Bootstrap subtheme?" contains a lot of details to get started with such sub-theme ...