In recent years, a new way of moving forward in Zenkutsu Dachi has been established. The more traditional approach has always been to pull the back leg as close as possible to the front leg and from there to move it forward to the front.
In the new system your legs are of minor importance. Instead, you body center and thus your weight shall be moved to the front in order to initiate the motion. This works best by bending the front knee through relaxation and utilizing the so produced energy to start the motion. Not the power of your back leg but the relaxation of your front leg leads to the motion. The advantage is a lot more speed and initial quickness than with the traditional system.
Our friends from the Karate Dojo WaKu in Tokyo have made an excellent explanatory video about this approach.