Composite construction is present on the majority of current aircraft - it's just the percentage that is getting higher. Most aircraft have a thin copper mesh embedded under the gel coat to provide the required lightning and electrostatic protection. Bonding straps or metal to metal connections to main structure provide the necessary continuity to ground for the panel. Static wicks are normally attached to some metallic structure as they require low bonding resistance.
All-composite aircraft such as the Starship have particular problems as they require all power returns to be brought back to grounding plates near the power supplies as there is no convenient metal structure. It will be interesting to see how Boeing gets over this on the 787 especially with the "all-electric aircraft" requirement for very large amounts of power.