We investigate the dynamics of the one-dimensional strongly repulsive spin-1/2 Bose-Hubbard model for filling ν≤1. While at ν=1 the system is a Hubbard-Mott insulator exhibiting dynamical properties of the Heisenberg ferromagnet, at ν<1 it is a ferromagnetic liquid with complex spin dynamics. We find that close to the insulator-liquid transition the system admits for a complete separation of spin and density degrees of freedom valid at all energy and momentum scales within the t-J approximation. This allows us to derive the propagator of transverse spin waves and the shape of the magnon peak in the dynamic spin structure factor.