Motorola has finally started rolling out Android 5.1 Lollipop to the Moto X 1st Gen (2013). India is one of the countries that is getting the new Android Lollipop goodness for the Moto X 2013, other countries include Australia, Hong Kong, Kuwait, Malaysia, Oman, Qatar, Saudi Arabia, Singapore, United Arab Emirates, Belgium, Italy, Netherlands and Spain.
The update is only available for retail units and not carrier branded/subsidized units. Moreover, the new Moto X 2014 (2nd-gen) is also getting Android 5.1 Lollipop update on both AT&T and US Cellular.
Source: David Schuster
