Sadly, these are a bit more annoying than super arts and Pandora activation. It will require you to equip whatever gem units you can to boost your cross meter, ideally by landing special moves if you can. Either way, the character you'll be using is Yoshimitsu. The move you'll be using is half-circle backwards + punch. This is the best meter bolstering move in the game and you'll need to do it for both Cross Assaults and Cross Arts.
Cross Assaults are activated by pressing quarter-circle back + Medium Punch and Kick. This should call in your partner and you'll see both characters on-screen attacking your foe. Be careful, if your characters are on opposite sides of the foe they'll auto-block most of your attacks making it difficult to finish them off in time.
If you'd like to switch characters to get two activated assaults rather than one, that should save you a few loading screens. This shouldn't be done until after your 300 wins, however.