It is the relative speed of the contact point of the ball and the strings that matters. The relative speed is composed of the ball velocity parallel to the strings, the rotational speed and direction of the ball at its contact with the strings, and the speed of the strings/racquet parallel to the ball. The relationship does not care whether the racquet speed is 0 or 100 mph; or whether the ball is 0 or 100; or any other combination.
If the ball impacts with backspin to the racquet, as it usually does, then the relative speed of ball to strings will be ball speed at the center of mass plus rotational speed at the bottom of the ball plus the racquet speed parallel to the ball, all considered as positive numbers. The relative speed will determine how long friction must act to slow the ball parallel to strings and change the spin in the opposite direction until the relative speed of the bottom the the ball and racquet equals zero, at which point the ball is said to bite the strings. It does this the same way whether the relative speed is caused by the ball alone (clamped racquet), racquet alone (stationary ball--e.g., serve), or a combination of each (groundstroke).
Friction duration just cares about the relative speeds. The magnitude of the friction force will indeed depend on whether the racquet is clamped or swinging or the ball stationary or moving, or a combination. But whether it is a big friction force or a small one, they both act until biting occurs, at which point the spin will be about the same, even though they take different amounts of time to achieve this. But a moving racquet does not add and special tangential force. That force is friction and friction will act until the ball bites. The fast moving racquet will increase the normal force which will increase the friction force, but, again, that will terminate upon biting.
When strings move out of position they store tangential elastic energy. Thus, we then have two tangential forces: friction and elastic. Friction will both reverse the spin of the ball and move the string, storing some otherwise wasted tangential energy. When the ball bites, the friction terminates but the elastic energy is still available to spin the ball. Friction again will contribute by determining the snap-back string's grip on the ball.
In fact, friction is again the main actor. As the string begins to snap back, that once again increases the relative speed of string to ball and friction will then again start pushing against this motion and spinning the ball at the same time. The elastic energy stored in the string is transferred to the ball via friction.