  In this paper, we introduce morphisms for matroids with coefficients (in the
sense of Baker and Bowler) and quiver matroids. We investigate their basic
properties, such as functoriality, duality, minors and cryptomorphic
characterizations in terms of vectors, circuits and bases (a.k.a.
Grassmann-Pl\"ucker functions). We generalize quiver matroids to quiver matroid
bundles and construct their moduli space, which is an $\mathbb{F}_1$-analogue
of a complex quiver Grassmannian. Eventually we introduce a suitable
interpretation of $\mathbb{F}_1$-points for these moduli spaces, so that in
'nice' cases their number is equal to the Euler characteristic of the
associated complex quiver Grassmannian.
