Oi-tsuki (literally: “lunge punch”), which we will call a pursuit punch, is a punch thrown while taking a step forward. In Aikidō, this attack is called a mune-tsuki ("middle punch").
Most untrained fighters will intuitively step with their punches, to “throw their weight” into them for extra power. Because of this, many of our self-defense techniques are defenses against pursuit punches. While the large, lunging movement (stepping into range) is the main source of power, all of the general rules for punching, and the specifics discussed in the overview, and in describing the front-foot punch and reverse punch still hold true.
From a front stance, or fighting stance, pull your rear leg up into a cat stance. In karate, everything must come in before it can go out; everything must be directed towards your centerline before it can be directed away from it. Then, push the unweighted leg forward, stepping into the other front stance, punching as you step. Your foot should trace a crescent-moon shape on the mat.
[video of pursuit punch from the front and from the side.]
When performing a pursuit punch, timing is everything. The punch must land at the exact moment that you lock yourself into your front stance. Attacking too early results in awkwardly pushing your opponent like a shopping cart. Attacking too late results in a weak punch which only uses arm strength. You must have good control of your upper and lower body, moving them simultaneously with precise timing. As such, pursuit punches are an excellent coordination drill.
Your head should not bob up and down as you step. Bobbing up and down is wasted motion, and the result of weak stances. Your head should remain in one plane through the entirety of a pursuit punch. Likewise, you should feel great strain in your thighs, if not, then your stances are too high.
A strong punch will not feel strong, because the power comes from the legs and core muscles, and not the arms. The arms are just along for the ride, whipping and snapping into place. The gauge of good technique is to look at the ends of the belt on your uniform. Strong techniques are derived from a snapping hip rotation, which causes the ends of your belt to flail around. A still belt is the hallmark of poor technique, so be sure to “whip the hip.” (This is what powers Mr. Miyagi's "drum punches".)