The wheel should not really move on its own with the bike in 1st gear and the clutch in, right?
NOT if you're on it.
Sounds like you may have too much play in your lever. Your play in between the lever and the perch should be approximately the thickness of a nickel.
I disagree. You have TOO LITTLE play.
Remove all the play and your clutch will NOT disengage; add TOO MUCH and your clutch will NOT engage.
Confirm - your clutch disengages w/ the lever FAR from the bars, right? Quite a few of the members have had their clutch cable break, pulling the ferrule (that goes into the lever) off, which is a direct result of having the disengage point FAR from the handlebars. Once you pull past the disengage point, (lever to the bars for instance as in waiting at a light) you're trying to pull the ferrule off.
I've adjusted quite a few Kawasakis to get the disengage point closer to the bars - your cable will last longer, and IF you need to modulate the clutch while riding (stuck, for instance), it's MUCH easier w/ the disengage point nearer to the bars.
Here's a pic of where mine is (I'm still on my original cable in BOTH my Vs, one at 51,xxx miles, the other at 47,935 kms!).