If you wish to move an item from one OS to another, it's simply a matter of copying the necessary files to their appropriate folder on the new computer.
For accounts, that's the keystore files under Accounts -> Backup -> Accounts.
For wallet contracts, that's the items in the application data folder.
It may be that the application data differs slightly between one OS to another. If your wallet contract still isn't showing up, click ADD WALLET, click IMPORT WALLET, and then paste the address of your wallet contract and import it.
As a note, while doing this you should always copy file and never delete any files until you are 100% certain you have 100% access to all accounts / wallet contracts.
This has the full breakdown of where the keystore files (accounts) and application data (wallet contracts) are stored on Windows, Mac, and Linux