In most Martial Arts system, you will see the practitioner step in, land, and then throw a punch.
In Jeet Kune Do we believe in moving the hand first. If the hand moves first it will get to the target first. If say the front foot moves first, it will be the first to land.
The hand moving first has the added benefit of allowing the punch to be less telegraphic. The reason behind this is, the smaller something is the harder it is to see coming. An opponent will be able to see my entire body coming towards him much easier than my and my much smaller hand.
Another added benefit of moving hand first, is not losing momentum after the initial push shuffle. When stepping, landing and then punching, all the momentum generated from upon the initial burst of acceleration is lost once both feet touch the ground. If the punch lands before the feet touch the groud then all that momentum is delivered into the punch thus generating more power. And at the end of the day that is what we want. How to maximize our punching power with our given skill set/body type.