You can transfer the game progress from your Mac to your new Mac manually by copying the folder with game saves.

Please follow the steps below:

1 - On your older computer, make sure you exited the game correctly last time. You can open the game again and then exit it again this way: Press the ESC button on your keyboard when you are in the City View, choose the option to exit the game.

2 - Then find the following folder with saves and copy it to your new computer:

%UserName%\Library\Containers\com.awem.cradleofempires.mac\Data\Library\Application Support\Awem\Cradle Of Empires\

NOTE: 

• %UserName% is your user name on the Mac.
• ~/Library folder is a hidden folder. To access it, first click the Go  in the menu and then press and hold the Alt key on your keyboard. You will see that the Library folder will appear. Then follow the path to your save as suggested above.

Important: if you updated your MacOS to Big Sur already, path to game saves will differ.

%UserName%\Library\Containers\Cradle Of Empires\Data\Library\Application Support\Awem\Cradle Of Empires\

3 - On your new computer, install the game and open it at least once so that a folder for saves can be generated.

4 - Exit the game correctly: Press the ESC button on your keyboard when you are in the City View, choose the option to exit the game.

5 - Now you need to replace the existing Cradle Of Empires folder with the Cradle Of Empires from your previous computer.