Charles,

I created a new document, then tried copying all styles. It got all of the headings, but not the Caption style.

First, the best way to get your styles into a new document is to create a template with those styles and create a new document based on the template. See Templates in Microsoft Word

The Caption style is a built-in style. Out of the box, it is based on the normal style. When you made your copy, did you copy the normal style as well as the Caption style? Did you copy them three times?

These styles are changed by style sets and by themes. The color is set as Accent 1 on my computer. Any chance your base document is using a different style set or theme than your target document?

What is the theme color palette in both documents, are they the same? You can check this by looking at the Modify Style dialog and clicking on the button for color. The top row is the Theme Colors for the theme in use with that style set and theme.

I would consider going into a copy of your base document and modifying the style to not be based on any other style. Then assign a real color to it, not a theme color, that is, one from the choices under the theme row. See http://www.gcflearnfree.org/word2010/18 for more on styles and themes.