Happened last week to my playing partner when his ball ended up under a tree.

During his backswing, he clips a branch and it ends up breaking then falling on top of his ball and making it move. While this was happening, he never stops his swing and proceeded to shank the ball. We had no clue if it was a penalty so we just moved on. What's the correct ruling? Thanks.

As long as it was clipped during the swing and not a practice swing.. There is no penalty..

What about causing the ball to move?

as long as it moved during a swing and not a practice swing, there shouldn't be a penalty.

It's a difficult ruling. The rules are not 100% clear on this matter. Because the players actions caused the ball to move, it could be penalized. Decision 14-5/1 (playing a moving ball) states:
Q- A players ball starts moving during his backswing and he strikes the ball while it is still moving. What is the ruling?
A- There is no penalty under rule 14-5 because the ball began to move after the player had begun his backswing.
However, if the player caused the ball to move, had addressed it or had removed a loose impediment within a club length of it, he incurred a penalty stroke - Rules 18-2a, b or c.

Since the action of his swing is directly attributable to the branch breaking, he would not normally incur a penalty stroke if he continued his swing however. Rule 13-2 states in part...." a player shall not improve...the area of his intended swing...except...in making a stroke or the backward movement of the club for a stroke. There is no penalty for breaking a branch in the act of making a stroke as long as he continued with, i.e. did not stop or discontinue the swing.
That's pretty clear - and probably where you are going wrong. The branch breaking as a result of the swing is not the issue. The branch subsequently moving the ball is. I would therefore say that it would be justified to invoke a penalty of one stroke. Tough call to make on this scenario. If the player stops to replace the ball to it's original position - he's penalized. If he continues with his swing - he's also penalized.

Rule 14-5 (Playing a Moving Ball) says "... When the ball begins to move only after the player has begun the stroke or the backward movement of his club for the stroke, he incurs no penalty under [14-5] for playing a moving ball, but he is not exempt from any penalty under [Rules 18-2a and 18-2b] ..."

Rule 18-2 assesses the player with a one stroke penalty for moving a ball at rest.

So, I think you take a penalty stroke under Rule 18-2. Since the ball moved after the backwards portion of the stroke had begun and the stroke was completed, you play the ball as it lies.

As noted previously, there is no penalty under 13-2.