You'll want a relaxed wrist and grip in most instances. Yes, the laws of physics do not care whether a wrist is moving or not when stick strikes ball, but holding the wrist stiffly can cause repetitive motion injuries from impact.
On the other hand, (pun intended) a subtly moving wrist can impart speed to the stick, providing distance and spin easily. There are six motions your wrist can make, I've sorted to 3 categories of 2 directions each:
1. ulnar deviation (down from wrist driving the pinky towards the forearm) and radial deviation (up from wrist driving the base of the thumb to the arm)
2. palmar flexion (straight forwards sending fingertips toward the smooth upper arm) and dorsal flexion (straight backwards moving the back of your hand to the hairy side of the forearm)
3. pronative (roll wrist forwards from the elbow of the arm on down) and supinative (roll backwards from the elbow)
Cup a billiard ball and toss it forward alongside your hip, ending with palm to the sky. You've naturally used palmer and dorsal flexion. Cup again, then rotate your forearm 90 degrees, placing the thumb and forefinger facing up, and toss again without releasing the ball. The weight of the ball will have your wrist perform ulnar and radial deviation instead.
Now lower the whole arm and your trunk to roll the ball along a pool table's felt and you have the pool stroking motion. "Tossing" the cuestick without actually letting go provides weight-induced deviation on backswing and followthrough (and for some, a tad of supinative or pronative roll, too).
Examine my hand on this exaggerated stroke and see the wrist motion, occuring easily if not forced. In this closeup and the previous photo, note the wrinkles in my wrist. That's pool!
