We develop an analytical microscopic theory to describe the polaron-polariton dispersion, formed by hybridizing excitons, photons, and phonons, and their coherent dynamics inside optical cavities. Starting from a microscopic light-matter Hamiltonian, we derive a simple analytical model by pursuing a non-perturbative treatment of the phonon and photon couplings to excitons. Within our theoretical framework, the phonons are treated as classical fields that are then quantized via the Floquet formalism. We show that, to a good approximation, the entire polaron-polariton system can be described using a band picture despite the phonons breaking translational symmetry. Our theory also sheds light on the long-lived coherent ballistic motion of exciton-polaritons with high excitonic character that propagate with group velocities lower than is expected from pure exciton-polariton bands, offering a microscopic explanation for these puzzling experimental observations.