Few days back Google Pixel's System Dump was leaked on the Internet and We found that the Smartphones comes with a new Google Dialer and Contact Application which can be easily Installed on Nexus 5X and Nexus 6P.
However, Developer kulankit97 (XDA) found a solution for Installing the new Google Dialer from Pixel on Motorola Smartphone running Android 6.0 Marshmallow. (Tested on Moto G4 Play, Moto G4 Plus and Moto G3)
You can follow the below Video Tutorial to Learn How to Install the Newest Google Dialer on Moto Phones.
How to Install New Google's Pixel Dialer App on Moto Smartphone-
Step 1. Download the MM... zip file, Google Dialer and Contacts App from Below.
Step 2. Boot to Recovery Mode and Flash that MM... Zip file.
Step 3. After Flashing the Zip file, Turn on your System.
Step 4. Go to File manager and Install other two Apk ( Google Dialer and Contacts ).
Step 5. Go to Settings > Apps > Gear Icon and Set Default Phone App to Second Phone App.
MM... zip file- Download Here
Google Phone- Download Here
Google Dialer - Download Here