  We study the moduli space of $d$-dimensional linear subspaces contained in a
fixed $(d+1)$-dimensional linear variety $X$, and its tropicalization. We prove
that these moduli spaces are linear subspaces themselves, and thus their
tropicalization is completely determined by their associated (valuated)
matroids. We show that these matroids can be interpreted as the matroid of
lines of the hyperplane arrangement corresponding to $X$, and generically are
equal to a Dilworth truncation of the free matroid. In this way, we can
describe combinatorially tropicalized Fano schemes and tropicalizations of
moduli spaces of stable maps of degree $1$ to a plane.
