  This paper is devoted to the construction of approximations of the propagator
associated with a semi-classical matrix-valued Schr\"odinger operator with
symbol presenting smooth eigenvalues crossings. Inspired by the approach of the
theoretical chemists Herman and Kluk who propagated continuous superpositions
of Gaussian wave-packets for scalar equations, we consider frozen and thawed
Gaussian initial value representations that incorporate classical transport and
branching processes along a hopping hypersurface. Based on the Gaussian
wave-packet framework, our result relies on an accurate analysis of the
solutions of the associated Schr\"odinger equation for data that are
vector-valued wave-packets. We prove that these solutions are asymptotic to
wavepackets at any order in terms of the semi-classical parameter.
