Can a kite take advantage of crosswinds to gain speed?
Why do you need a kite? A glider is much better suited to the task.

Again, nature has invented this long ago. Albatrosses use a technique called dynamic soaring to cover wide distances over water, using the windshear close to the surface.

Humans have tried to imitate this with gliders, albeit with limited success. The few who tried it could indeed stay in the air by harvesting energy from windshear, but the dynamic loads (up to 4 g) were too much to do this over an extended time.