A note to judges:
This is not the place or time to try and make you read the rule book. You've read it. You know what it says (although a refresher may help many of us.) I just wanted to talk to you about how to place weight on a hound actions. Good or bad. It's vitally important for you to determine what effect the hounds actions are having on the race.
It would indeed be short sighted to quickly pick up a top performing hound with things that don't detriment your race. Like wise you should never credit a hound with score accomplished with some type of gambling type action. Running hit and miss or bounding off at the checks should never be tolerated. Dogs with these faults should be eliminated as soon as you determine the action is indeed a habit or method the hound uses. A single time of being out of place is not a habit! The other thing I see is the advancement of dogs who do not have quality performances. Simply not being faulty, and picking a check or two should not guarantee advancement into second series. Judges, you should insure every dog you bring back to second series can indeed run a rabbit. Even if that means running the pack down to one dog and watching him run for a few minutes. Most competitors will understand this and look forward to their hound getting a chance to display his qualities. If the dogs is unable to run satisfactorily by himself he should not be advanced. The handler will understand if he is a hounds man.

Here are some example I have seen or heard of dogs being eliminated for that don't seem to hurt the race: In these example Dog B will get picked up.When a dog A strikes the track of jumps the rabbit, dog B tongues with excitement as he harks to dogs A's location. Dog B gets picked up four being mouthy.
At a check in and around a small briar thicket and all the dogs are searching. Dog A picks up the check and drives into the briar thicket, Dog B is already forward and off to the side of the thicket searching and meets dog A as he emerges out the other side of the thicket. Dog B is eliminated for skirting.
The pack is driving and a dog back in the pack pops out off the track. He stays out for a couple seconds and them pops back in the pack in roughly the same relative position. The dog get's eliminated for "getting out."
The dogs are running good with dogs A and B both doing the majority of the scoring. A long check arises and after 2 minutes dog B opens exactly at the point of loss as he hunts back across the check area. Dog B gets picked up for being mouthy.
Now these are things I pulled off the top of my head but I want to point out these things probably were not hindering the race.

Here a some reasons that should get a dog eliminated: In these example dog B will get picked up and assuming first series in a reasonably large trial.The pack is driving hard. Dogs A and B are in the front racing. All of a sudden dogs start pulling up as dog B continues tonguing another 50 feet. The dogs work back where dog A finds the loss and the pack is off. Dog B races to the front in an attempt to run the lead. Again, dogs start pulling up and we find Dog A and dog B 50 feet past where the rabbit turned. Dog B should be eliminated for racing and over running.
The pack is running smoothly with little down time. Naturally there are checks but most are solved in 5 - 10 seconds. A hard check arises and 20 seconds later we here dog B 100 feet out with a rabbit track. The dogs hark to him and the race is on. More good running with several different dogs running the front and solving checks. Another long check and in less than a minute we again here dog B over the hill with a rabbit. Dog B should be eliminated for lack of patience and bounding off.
The pack is running and come to a check. Dog B indicates he solved the check by tonguing and pulling all the dogs to him. After they get to him he stops barking and continues hunting. The check is finally solved and the dogs are off again. Dog B repeats the same process of tonguing where no rabbit can be found. Dog B should be eliminated.
You've been struggling with this pack and have picked up several faulty dogs. You are down to three dogs and are not running very well. Dog C starts giving extra mouth and you pick him up. Now you are down to dogs A and B. Dog A is getting most of the work and dog B is really nor scoring. Dog B should be eliminated.

There are so many variable it is nearly impossible to simply put every scenario on paper. You really need to follow a lot of dogs. Develop an eye for what hurts a race and what has little effect. Let the little thing decide between successful performers. You shouldn't start a race looking for reason to "pick up" dogs. Look for good things the dogs do to contribute to the race. You will have plenty of time to eliminate dogs as your scoring goes along.