When to put new ball into play after HR?

[DNTXUM P]

Speaking strictly NCAA - in most cases after a home run, the catcher is already in the circle with the pitcher. If so, I throw the ball to the pitcher or catcher - whoever is looking for it - after all runners have crossed home plate. If the catcher is still at home plate, I give her the ball and allow her to throw/bring the ball to the pitcher even though Emily wants the PU to throw all balls to the pitcher, not give them to the catcher. Also for NCAA - they want you throwing the ball overhand, not underhand - shows more strength.