  We fully investigate the symmetry breaking patterns occurring upon creation
of composite non-Abelian strings: vortex strings in non-Abelian theories where
different sets of colours have different amounts of flux. After spontaneous
symmetry breaking, there remains some internal colour degrees of freedom
attached to these objects, which we argue must exist in a Flag manifold, a more
general kind of projective space than both $\mathbb{CP}(N)$ and the
Grassmannian manifold. These strings are expected to be BPS, since its
constituents are. We demonstrate that this is true and construct a low-energy
effective action for the fluctuations of the internal Flag moduli, which we
then re-write it in two different ways for the dynamics of these degrees of
freedom: a gauged linear sigma model with auxiliary fields and a non-linear
sigma model with an explicit target space metric for the Flag Manifolds, both
of which $\mathcal{N}=(2,2)$ supersymmetric. We finish by performing some
groundwork analysis of the resulting theory.
