bb100 R Batman and Catwoman
Team: Batman Ally
Range: 3
Points: 142
Keywords: Detective, Gotham City

8 10 17 3

8 9 17 2

8 9 17 2

8 9 16 2

7 8 16 2

7 9 16 3

7 9 15 2

7 8 15 2

KO KO KO KO

KO KO KO KO

KO KO KO KO

KO KO KO KO

(Special) I Already Stole the Real One: Once at the beginning of the game, after objects are placed, you may remove an object placed by an opponent from the map and place a standard light object in the same square.

(Speed) Double Pounce : Batman and Catwoman can use Leap/Climb. When Batman and Catwoman are given a move action, after the movement is complete Batman and Catwoman may use the Duo Attack ability as a free action.

(Attack) Summon the Bat or Set the Trap : Give Batman and Catwoman a power action and choose target character within 6 squares who possesses the Batman Ally or the Batman Enemy Team ability (line of fire is not required). Place the target character in an unoccupied square adjacent to Batman and Catwoman. If the target is friendly, it can't be given an action until your next turn.

(Defense) Romantic Tension : Batman and Catwoman can use Combat Reflexes and Willpower.

And:
Grapnel Gun
Resource

Give this character a move action; it can use Improved Movement: [ELEVATED] and the Carry ability for this action. If the square the character ends its movement in is of a higher elevation than the square it began the movement, it may be given a close combat action as a free action after actions resolve.

Double Pounce says "When Batman and Catwoman are given a move action, after the movement is complete Batman and Catwoman may use the Duo Attack ability as a free action."

What I want to know is after they move, since it says the attacking part is a free action, could I use Outwit, Perplex, etc. before the attack? I've got the very same question for the Grapnel Gun, as well.

No to both.

In case of Batman & Wonder Woman the Duo Attack ability is activated while the action is still resolving.

For the Grapnel Gun see page for in the CRS: