You can, provided that your devices run the same OS. In other cases, it’s currently not possible — there's no way to transfer progress to a different OS.
First of all, on a device you want your Little Big Robots profile transferred from make sure that this account is connected to Game Center on iOS or Google Play Games on Android: check the game menu and make sure that the your progress is saved.
On the device you’d like to transfer your Little Big Robots account to:
- Log in with the same Game Center/Google Play Games account your Little Big Robots game account is connected to on your device;
- Launch Little Big Robots;
- Load your game progress when suggested.
⚠ NOTE: Only one Little Big Robots game account can be connected to a single game service account (Game Center/Google Play games).