This won't always work. When a copy of a page like you describe interacts with the parent site, the latter can tell that the page isn't local, and they might care. I had a case like that on my site some years ago -- people were trying to hack my guestbook page by downloading it and modifying it. I finally stopped that activity by detecting the origin IP and rejecting any activity that wasn't local.
Thanks for the comment. Yes this won't work for sites that are heavily dependent on interaction with the server. In my experience it's better for working with static sites such as blogs, news sites, etc.