Dynamic Wireless Power Transfer (DWPT) is a novel technology that allows charging an electric vehicle while driving thanks to a dedicated road infrastructure. DWPT's capabilities in automatically establishing charging sessions and billing without users' intervention make it prone to cybersecurity attacks. Hence, security is essential in preventing fraud, impersonation, and user tracking. To this aim, researchers proposed different solutions for authenticating users. However, recent advancements in quantum computing jeopardize classical public key cryptography, making currently existing solutions in DWPT authentication nonviable. To avoid the resource burden imposed by technology upgrades, it is essential to develop post-quantum-resistant solutions. In this paper, we propose DynamiQS, the first post-quantum secure authentication protocol for dynamic wireless charging. DynamiQS is privacy-preserving and secure against attacks on the DWPT. We leverage an Identity-Based Encryption with Lattices in the Ring Learning With Error framework. Furthermore, we show the possibility of using DynamiQS in a real environment, leveraging the results of cryptographic computation on real constrained devices and simulations. DynamiQS reaches a total time cost of around 281 ms, which is practicable in dynamic charging settings (car and charging infrastructure).@en