In the aftermath of Apple’s announcements earlier today, it released an update to iTunes with a new feature described as rent once, watch anywhere. The release notes for iTunes 12.6, which is available as a free upgrade on the Mac App Store, say:
Now you can enjoy your iTunes movie rentals across your devices with iOS 10.3 or tvOS 10.2.
With this new feature, you should be able to start a movie rental on an iOS device and finish it at home on an Apple TV for example, which is a welcome change to what was an overly inflexible system.
iOS 10.3 and tvOS 10.2 are currently in beta but are expected to be released soon, at which time this new feature will be available to everyone who upgrades to those versions of the OSes.