This technique has a name: pump(ing) side kick. It's used to advance by sliding and meant to a) prevent the opponent from directly attacking (functioning almost as a block) and b) have the leg up and chambered to deliver a side kick. The power part of this is not the actual pump kick, it's the final part of the kick, the one that actually makes contact. At that point, the technique requires the hip to rotate and open (I tell my juniors to point their butts at the target), locking out the knee and thrusting the heel through the target. This is called "sticking" the kick. The power comes from the rotation of the hip and the push against the floor (that's poorly worded, but WTH) of the base foot. Sport karate fighters (and Chelsey Nash is one of the top women) in NASKA/ISKA are supposed to use light contact (little or no penetration) to the face and head, and light-moderate contact to the body, but if the sticking portion of the kick isn't pulled by softening the knee, you can break ribs very easily.
In essence, it's almost a distraction technique, allowing you to get set up for a good, strong side kick with hip rotation and a solid base. Of course, there are a lot of people who do it incorrectly, kicking with their bellies pointing forward, which is awkward and creates an ugly, weak kick.
That being said, I fight in this style, and I only pump once or twice, not over and over--just enough to close distance quickly and set up the offensive side kick. I love to fight people who pump a lot, because you can use your angles (someone up there said "circle them") to get around the leg, and it's hard to change direction quickly with one foot in the air. I also tend to go for economy of motion more--I'm old.
I will also say that a front leg side kick, delivered correctly, is hella powerful. Just check out old videos of Bill Wallace's full contact fights. I've kicked women who outweighed me by 50 pounds right out of the ring--this is Texas. A point in Texas is a felony anywhere else.