Execution:

The pull up starts with the athlete hanging from the rig, hands just outside shoulder width from a dead hang. The pullup is initiated with the shoulders, the athlete performs a kipping swing, tapping between hollow and arch.
Using the momentum of the kip by pushing down with the lats in the hollow position the athlete drives the hips towards the bar and then pulls the chin over the bar, completing the rep when the chin is higher than the bar.
To link the next rep the athlete pushes away slightly at the top of the rep (where they are in a tight hollow position) and allows the body to follow through into a full arch underneath the rig.

Purpose:

The purpose of doing a kipping pull up vs a strict pull up is to increase the number of pull up repetitions the athlete can perform. It also helps to develop a strong core and motor control, having to strengthen the key gymnastics movements of hollow and arch.

Variations:

Strict pull ups (overhand, underhand or mixed grip), chest to bar pull ups (where the finish point is chest physically making contact with the bar below the clavicle) and butterfly pull ups.

Programming:

Kipping pull ups are frequently programmed in wods, although not specifically as “kipping” any variation may be used. The kipping movement is also used on the rings as a muscle up development tool. Other variations of the pull up, like strict, chest to bar and butterfly pull ups are often used in strength and skill sessions at the start of a class to help develop strength or work of movement patterns.