Yes downloading is available for movies and tv shows. When you pull up a movie and purchase it the option to download should be right in the three quick options tab: pin to start-download-share. Also if you are having the error code or failure to obtain license message pop up, go to
www.xbox.com and sign in. Go to entertainment and do the one time transfer thing on the desktop/tablet first and then you'll be able to purchase, stream, and download tv shows and movies directly on your phone. So glad this feature is finally available to us.
Edit: I just tried to download a purchased movie and still got the error message so looks like downloads are only working for the tv shows.