Personally I don't like the term as it always makes me wonder if there's a perceived hierarchy to feminism, that intersectional feminism is somehow superior. The term certainly suggests that unless it's intersectional, feminism isn't inclusive of all women which I strongly disagree with.
I think it's fair to say that many who identify as IF see themselves as superior, but that's not the same thing, of course.
IMO feminism (as you say) crosses lines of race, class, education and nationality, but cannot, by definition, cross sex lines, as doing so would mean it was not feminism but something different (the 'something' being dependent on the objectives of the group). So white British feminists would be supportive of black american women's rights, but if they supported white British men (as well they may) it would not be driven by feminism, but by politics or something else.