In a food processor, combine the basil, dill, chives, garlic, parmesan, lemon juice, and a pinch of salt and pepper. With the food processor running, slowly stream in the olive oil until emulsified.
Drain the mozzarella balls and pour into a bowl. Toss with a few spoonfuls of the pesto. Let sit while you prepare the rest of the skewer ingredients.
Halve each melon and scoop out the stringy flesh and seeds. Use a melon baller to twist neat little spheres of melon. Set aside in a bowl. You may end up with more than you need for the skewers; miscellaneous snacking, ahoy!
Halve the cucumber and trim off the ends. Use a mandolin to produce thin strips.
Place a mozzarella ball in a big basil leaf and thread onto the skewer. Follow with a melon ball, a cucumber strip, another mozzarella ball, melon, cucumber, and finish with a mozzarella ball. Repeat this pattern with remaining ingredients, or customize to your own liking! Serve with remaining pesto alongside.