Using child components props harmful?

DocumentTab.js

class DocumentTab extends React.Component {

  static propTypes = Tab.propTypes;
  static defaultProps = Object.assign({}, Tab.defaultProps, {
    title: "Document Builds",
  });

  render() {
    const { document_builds, ...props } = this.props;
    return (
      <Tab {...props}>
        {(document_builds || []).map(document_build => (
          <DocumentBuild key="..."
            document_build={document_build}
          />
        ))}
      </Tab>
    );
  }

}

Page.js

<Tabs>
  <Tab title="Normal Tab">
     <div>Some Content</div>
  </Tab>
  <DocumentTab document_builds={document_builds} />
</Tabs>

Tabs.js

// Not exported - Shows tab links
const TabNav = (props) => {
  //...
  return (
    <ul className={tabNavClassName} role='tablist'>
      {
        React.Children.map(nonNullTabs, tab => (
          <TabItem
            {...tab.props} // <-- This is the problem - when using DocumentTab, tab.props has no title
          />
        ))
      }
    </ul>
  );
};

export const Tab = (props) => {
  const { className, eventKey, activeKey, children } = props;
  return (
    <TabContent className={className} active={eventKey === activeKey}>
      { children }
    </TabContent>
  );
};

Tab.propTypes = {
  title: PropTypes.string,
};

TabItem passing tab.props is the problem. My wrapper doesn't get title passed as a prop, and the defaultProp doesn't seem to be passed when props is accessed from the outside.