copyStyles.js

function copyStyles(sourceDoc, targetDoc) {
  Array.from(sourceDoc.styleSheets).forEach(styleSheet => {
    if (styleSheet.cssRules) { // for <style> elements
      const newStyleEl = sourceDoc.createElement('style');

      Array.from(styleSheet.cssRules).forEach(cssRule => {
        // write the text of each rule into the body of the style element
        newStyleEl.appendChild(sourceDoc.createTextNode(cssRule.cssText));
      });

      targetDoc.head.appendChild(newStyleEl);
    } else if (styleSheet.href) { // for <link> elements loading CSS from a URL
      const newLinkEl = sourceDoc.createElement('link');

      newLinkEl.rel = 'stylesheet';
      newLinkEl.href = styleSheet.href;
      targetDoc.head.appendChild(newLinkEl);
    }
  });
}

From Chrome 64 using .cssRules on sheets from other domains will throw, so it's important to first check if the sheet is external, then if the <style> element has any CSS rules.

function copyStyles(sourceDoc, targetDoc) {
  Array.from(sourceDoc.styleSheets).forEach(styleSheet => {
    if (styleSheet.href) {
      // for <link> elements loading CSS from a URL
      ...
    } else if (styleSheet.cssRules && styleSheet.cssRules.length > 0) {
      // for <style> elements
      ...
    }
  });
}