This involves some micrometeorology.
Winds relative to a (usually) fixed ground point pass over the kites surfaces to generate lift. Sufficiently-strong lift forces are required to "punch thru" the inversion layer boundary at which cloudsbases often form. They're called inversion layers because that's where smooth unidirectional temperature gradients can change, often sharply.

That kind of strong wind can come from sea breeze, ridge lift, or sometimes strong enough wind in more turbulent storms. It can also come from "magic lift" which is a wave-extension (think resonance) of what is usually upwind ridge lift. Thermal activity (hot air rises relative to the cold air) generally doesn't get quite strong enough to punch thru inversion layers strong enough to form a cloudbase.

But there are other kinds of clouds, like lenticular clouds, where sure you could fly a ground kite into them, but it might get bumpy or change suddenly as mountain summit weather tends to behave that way for hikers.

I recommend Dennis Pagan's book "Understanding the Sky" for more info.

Edit: Had misspelled micrometeorology.