The problem: The horse begins stop correctly, but does not stay in the slide. He picks hind legs up and puts them down again like a stone skipping on water.
Why does this problem happen? Probably the number one reason a horse skips in the slide is the rider. If the rider's timing is not good or unsure, the horse's timing will not be either. The horse feels "stop, don't stop, stop, don't stop, etc" and so that's how he slides – in the ground, out of the ground, etc. If the rider gets ahead of his horse and then throws his shoulders back with one sudden movement for the stop, the horse may skip as well.
Skipping may occur if the horse is not confident enough to commit to the ground or if he is not capable of staying in the ground when the stop is approached with great speed. (I once saw a "longest slide" contest at a reining show and many of the horses skipped because the riders, trying to win the contest, had asked with maximum speed.)
How to correct this problem: If I, the rider, am causing the problem, I can correct that. I don't want to ride forward in the rundown and suddenly throw my shoulders back when I want to stop. I do want my shoulders to be behind the motion in the rundown and when I prepare to stop so my horse is running out ahead of me. If I ride every stride, getting deeper and deeper in the saddle in the rundown, then all I need to do is push into the stirrups, sit down, lock my back and say "whoa".
If I can find nothing wrong with my position and my horse still skips, I must slow things down and review his basic training. All the basic requirements for a correct rundown and a correct approach to the stop come into play – straight and soft in the rundown, wants to stop, will stop without rein pressure, will accept rein pressure if needed – to achieve a sliding stop with no skipping. If these things are not solid, I school my horse until they are. Then my job, as the rider, is to deliver a very clear, consistent message to my horse for the stop. If I believe, he will.
Another exercise that may help is fencing (See explanation of fencing in Stop Fix #1: Straighten the Rundown) because it can teach the horse to break at the loin and become more committed to the stop.