Hi Thomas:
No. As I said on your other thread earlier today, variabled are defined document-wide and we can't tie a variable to a section but, now that we can see your image, there is a way to automate this.
And to adrdess a recent question, the term running head refers to the repeating content in both the top margin (aka header) and in bottom margin (aka footer).
Assuming you want the word General in the running head under Section 1, add it to the body page, create a paragraph style for it and then add a Running Head (Paragraph style) variable to your parent page to pull in the content, based on the assigned paragraph style. InDesign will pull the content from the page (in this case, General) and will repeat the content in the header until it changes at the beginning of the next section. Automatically. This feature is used to pull chapter names, chapter numbers and subheads into the running heads, which is exactly what you are looking for.
~Barb
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