Currently this can be done by using Google Sign-In or Facebook Login in the game settings. Tap on the cogwheel in the top right of the Mansion screen to access these features in your existing game.
Connecting to Google Sign-In or Facebook Login means that your game progress will be saved to cloud storage. If you then wish to continue on another device, just install Merge Mansion on the new device and sign in to Google Sign-In or Facebook Login on that device.
Upon signing in with Google Sign-In or Facebook Login you will be able to continue playing Merge Mansion on the new device.
We highly recommend you connect to save your progress, however, If you do not wish to use Google Sign-In or Facebook Login and you are changing phones or planning on resetting your device, you can ask Support to transfer your game progress over to the new device by contacting us from within Merge Mansion.
To initiate the transfer, you need to provide the Player ID from the game you want to move and the Player ID from your newly installed game. The Player ID is attached to the messages you send to Support, so the easiest way to provide it is simply sending a message from both the old game and the new game, asking for transfer. A screenshot of your Settings menu will also show the ID.