In karate body dynamics and inside muscle action are the main sources of power (of course, controlled my the mind, so we could say the mind is the main source of energy).
Body shifting is one of the 6 body dynamics. Shifting is used to produce force as well as to adjust distance, and it is generally used to apply techniques in longer distance.
We use straight shifting, such as in case of Oi Zuki, or side shifting, such as in Yoko Enpi Ate (side elbow smash), and kizami Zuki in longer space.
Yori Ashi, Tsugi Ashi, Fumi Ashi can be used as means of shifting.
In the case of Oi Zuki for example, pure straight shifting should be used, introducing any rotation cause energy to split, since we want straight energy line.
In some cases shifting and rotation are combined, and shifting is mainly for purpose of distance, such as when stepping with Soto Uke or Age Uke, shifting does not produce force to the side or upward, therefore rotation is main energy.
Or when executing Mae Geri with back leg (stepping forward), main body action is pendulum action (which is rotation around an axis parallel to floor), shifting is not optimal for the direction of the kick, therefore is secondary, and used to allow transfer of momentum forward and setting optimal angle of support leg to body center to kick for energy transfer.
Or think of Yori Ashi (sliding) with Gyaku Zuki, first we shift and at last moment, as the front foot enter the floor, use rotation which is the main power.
In the above case, straight line energy is changing and adding into rotational energy, but shifting is secondary, since shifting needs big space to produce sufficient force.
If rotation starts as we start shifting, rotation will take place over long period and will be slow and ineffective, and in addition, our body center will be exposed too soon.
An exception is when gyaku zuki is executed in short space, than shifting and rotation starts together.
Shifting needs more space to produce big force, think of trying to break through a door, you need big distance to be able to increase enough force to break a door, it can’t be done from little space.
Or think of hitting a baseball or golf ball, no one shifts (runs sideways) in order to hit the ball, but rather rotation is being used.
In shorter space rotation is much more powerful.
It is a common problem to see someone shifts too much and having no space for rotation while executing gyaku zuki, which results in weak technique, but also exposing chance since shifting is slower than rotation.
Attention need to be placed on correct application of body shifting principles such as keeping the weight in between feet, and foot first, body center second and than technique, in order to effectively apply the points discussed above.
I leave that for another discussion.