Goal kick
 

My understanding of a goal kick is that it must be inside the lined area in front of the goal and the ball must be stationary. It is permissable to move the ball from one side to the other but it must be stationary before being kicked. The resolution to this would be a re-kick if the ball was still moving when struck by the defending team. If the defending team had a player that regularly moved the ball from one side or the other and kicked the ball b-4 it was stopped, how many times would you guys allow this before you Y-card for P-I? The level of play is U-13 classic so the kid should know how to react to referee instuctions.

The Laws of the Game consider a goal kick to be a free kick, so yes, the ball must be stationary when kicked. If it is not, the kick shall be retaken as you have stated.
To be properly placed the ball does not have to be inside the lines, but only has to be partially breaking the vertical plane of the outermost edge of the line. The ball can be on the line or resting just outside of it as long as part of the ball breaks the imaginery vertical plane.
Under the LOTG the ball may be placed and then moved. However, the player/team shall not waste time. This is up to the discretion of the referee. (Under HS rules, once placed the ball may not be moved -- that is deemed time wasting.) Technically, in USSF the caution should be given for Unsporting Behavior (time wasting) or Delaying the Restart because per USSF instruction PI only applies to Laws 12 and 14. (However, there is a sentence in the ATR that gives "Fails to start or restart play properly or promptly, having previously been warned" as an example of PI. Although I don't personally see how that sentence fits with the other as this is clearly Law 13. Chalk it up to one of the many conflicts in the ATR.)
Whether I caution or not at the U13 level would depend upon the skill of the players and the event. Is this a regional tournament, Super-Y league, a state cup, or local league play? I would definitely have a word with the player the first time it happened and inform him that the ball must stop before he kicks it.

PS I will NOT give a player a second caution for this.

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I would only card this moving from side to side if I considered it a time wasting tactic. Then I would write it up as UC rather than PI. That's probably just a matter of choice for each official, though. It sounds like more of a teaching opportunity to me. Even @ U-13 they can get so excited that they don't let it fully come to rest.

pls remeber to warn about these minor infractions before taking harsher measures. also, consider use of trifling.

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Trifling would be a good choice, imo, too.