see also: https://indesign.uservoice.com/forums/601021-adobe-indesign-feature-requests/suggestions/45188980-would-like-to-be-able-to-apply-formatting-to-text

The particular case you mention, if I am understanding it correctly, can be done with variables. My assumption is that you have the chapter number and title on one line in the book as well as the running heads. If that isn't the case, just having a separate variable for each paragraph style and placing those both in the running heads would work. Multiple variables can be used side by side and uniquely formatted.

If those two elements are in one line in the book, this is how I would deal with that:
First, create two character styles that don't actually need to do anything (one for chapter number and one for chapter title). Then, in the paragraph style you are using for that element, create a nested style that is applied through the first colon (using chapter number character style) and then create a second nested style applied through an "end nested style character" (using the chapter title character style), which will just run through the rest of the paragraph since that character isn't there. Then create 2 "Running Header (character style)" variables, one for each character style you created. Those variables can then be used side by side in the running heads and have different local formatting applied.

Hope that helps. I apologize if I misunderstood the issue.

Meanwhile, you could try using two frames on the master page, one for each variable.
The current problem with all text variables is that what's actually inserted in the frame is a single hidden character that references the text in the document, not the text itself. Being a single character, it can't have two character styles applied to diferent parts, nested styles don't apply, and GREP styles don't see text, just that hidden character. That's also why text variables can only sit on a single line, and if that is too long for the frame it gets squished.
I don't know what the engineering would have to be to change this behavior, but I'm guessing it's not trivial. For the moment, these problems are inherent in the basic architecture of the function. Hopefully, understanding the limitations might help with ideas for working around them.

Character styles carrying over to text variables would be really helpful.

Yes, please add this feature.

Would love to see this thanks.

Allowing character styles to carry over would be a big help.

Yes, please allow ialics (etc) to carry over to text variables

YES! Making multiple masters for chapters with itals in the title is annoying.