  We investigate the dynamics of the one-dimensional strongly repulsive
spin-1/2 Bose-Hubbard model for filling $\nu\le1.$ While at $\nu=1$ the system
is a Hubbard-Mott insulator exhibiting dynamical properties of the Heisenberg
ferromagnet, at $\nu<1$ 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 {\it 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.
