Countering mobile shore-based anti-ship missiles

How do modern navies deal with mobile shore-based anti-ship missiles?

They try to find and disrupt the kill chain or destroy the launcher before it can fire.

The compact answer focuses on the targeting system rather than only the missile itself. A naval force must disrupt the sensors, communications, and targeting links that allow a mobile launcher to engage, while also seeking the launcher directly. This is the core operational problem of fighting inside a modern missile threat zone.