Dungeon, Inc. uses either iCloud for iOS devices or Google Play Games for Android devices to save your progress to the cloud.
This means that you should be able to restore your progress if you reinstall the game, or upgrade to a new device.
To access the cloud save system in-game please tap on either the Game Center or Play Games icon at the top left of the screen, to the left of the Settings icon.
The cloud save systems is specific to their respective platforms. As such, transferring progress from iCloud to Play Games and vice versa is currently not supported.