The problem: The horse resists the rider's request to increase speed into the stop.
Why does this problem happen? Unless there is some physical reason the horse cannot accelerate to the stop, the only reason he doesn't is because he doesn't want to and doesn't respond when his rider asks. This happens because the horse has not been taught to accelerate when the rider asks or possibly because he has been stopped hard too much (he is sour).
How to correct this problem: If my horse does not accelerate in the rundown, the stop will be compromised. His stride will flatten and his weight will be more on the front, which means he will hit the ground hard when he stops. Although I don't want him running like a race horse (he might be thinking more of running than of stopping and ignore me), he must have some acceleration so his shoulders are elevated and his hind legs are well under him. In this position, he is capable of a nice long slide.
Exercise: Since the key to acceleration is to increase speed just a little every two or three strides in the rundown, I want to school my horse that way so that he responds when I ask in the show pen. One of the best ways to do this is to lope lots of straight lines with no stops in training sessions, sometimes asking for accelerated speed, sometimes not. I want my horse to begin to accelerate just a few strides into the rundown but I make sure he is collected and straight before I ask. Then I start riding deeper and deeper in the saddle all the way down the pen and let him run. (This is important – I must allow my horse to run if I am asking him to run!) If he does not increase speed with my body rhythm, I "cluck" to him and back it up with a bump with both legs if he does not respond to the "cluck". Of course I don't want my horse simply running with no collection and his head in the air. If that happens I will fix that first, asking him to give to the bridle and drive with his hind quarters (basic collection). If my horse has been well schooled, I can accomplish that in the first few strides of the rundown.
If I am consistent about this program, it will only take a "cluck" to remind him in the show pen. The object is for him to 1. Wait for me to ask and 2. Believe me when I ask.
Note: Every horse has an optimum acceleration at which he can stop. It's important to know how fast my horse should be running for a smooth, controlled stop. If the horse is not strong enough to stop from a powerful, fast rundown, asking him to do so will surely result in a trashed stop.
A good seat is important to a controlled, accelerated rundown. If I sit on "on my pockets" (not forward), with my shoulders squared and my rein hand forward riding every stride deeper and deeper, I will not only encourage a great rundown but will also be in a great position for the stop.
Note: One thing I see happen is the rider waiting too long to ask for acceleration into the stop. Half way down the pen is too far! As soon as my horse is collected and straight I ask him to increase speed – gradually! (See Stop Fix #3: Correct Break and Run Rundown if increasing speed gradually is a problem).