[trefn]

I wonder if you could make the kite itself lighter than air to avoid issues where the wind dies down? E.g. a kite shaped airship.

[usrusr]

There are no issues, at least as long as you can supply power to the base station: just make the motor/generator reel in the kite, that causes plenty of movement relative to the stationary air to keep it afloat and controllable right down to the reefing mechanism.