Sounds like you're describing throttle lag, or slow throttle response.
Slide should move freely, up and down. Diaphragm intact, vacuum port clear.
Don't want to get too involved here ("Care and Feeding of the CVK40" goes far enough), but . . . twisting the throttle grip opens an air valve into the carburetor venturi; the venturi vacuum, piped above the diaphragm, raises the slide and needle. Unless the slide, diaphragm, and needle are all o.k., a temporarily air-rich mixture results, sometimes resulting in the stumble described.
That said, a CV carburetor has an inherent delay between twisting the throttle grip and raising the slide and needle (feeding fuel into the venturi).
Thanks for your answer Damocles! I didnt really explain the problem that good, the bike actually stalls over 4k or so (sometimes for couple seconds), it seems like it would turn off if I didnt play with the throttle or change gear. If i kept throttle in the same position the stall would continue and probably it would stop.
What is interesting is that it rides perfectly until the temp gauge gets to about half or more. So I can ride normally, like 20 mins or more with great throttle response up to 7k, and when it gets really hot these problems start and happen almost every time i open the throttle harder, in higher revs (and they are worse up hill).
Only way i can ride it then is below 4k and not opening throttle all the way. And this can go on for hours, it rides great actually but you can just cruise.
So I am thinking it is something temperature specific or perhapse a problem that arises after riding aggresively for some time - not connected to temp??
You mention the vacuum port, could you please explain more? There is a port on top of the carb but it has a cap on it on my klr600, or do you mean something else?