We discuss in this talk the construction of initial value representations of solutions of systems of semi-classical equations given by a time-dependent matrix-valued Hamiltonian. Such systems could arise in mathematical relativity from a linearization process. These initial value representations are inspired by the approach of the theoretical chemists Herman and Kluk who propagated continuous superpositions of Gaussian wave-packets for scalar equations. We explain how to generalize their construction to systems, including those presenting (smooth) eigenvalue crossings by incorporating to classical transport a branching process along a hopping hypersurface. We also explain how these propagators give an algorithmic approximated description of the propagator.