OK I managed to move my PHS QT/wallet from Windows to Linux now.
In case someone faces the same challenge starting with Linux, here is what I had to do:

• Launch the wallet in windows, go to the console, and run the command
• dumpprivkey

• Do this for every PHS address in the wallet and copy the private key somewhere
• Install Linux QT and sync the blockchain, go to the console, and run the command
• importprivkey

• Do this for every PHS private key
• Then go to file (main QT window), backup wallet
• For some reason the wallet was saved in the QT folder, not the .Philosopherstone one
• Name the backup wallet.dat and move to .Philosopherstone hidden folder (needs admin permission)

That's it! Enjoy the staking