How soon do you give F1/F2 a new ball after an HR (over the fence)??

I wait until the batter/runner touches home and then toss a new ball to the pitcher.

If there are several runner then I stay around the plate to make sure the sportsmanship issues are adhered to.

If there are no runners, while BR is running I will usually go hand or toss a new ball to F1 while BR is on his way to second or third.

Then I am able to watch him touch 3B and Home etc.,

Thanks
David

Note, tornado (and this may be where you're going with this). The ball you throw to the pitcher, if you throw it early, is NOT a live ball for purposes of any appeals until the original batter enters his dugout and YOU make the ball live. You may feel it facilitates gameplay to give the pitcher or catcher A ball early... but it is not THE ball until much later.