  Using the notion of a valuation into the semifield of piecewise linear
functions, we give a classification of torus equivariant flat families of
finite type over a toric variety base, by certain piecewise linear maps between
fans. As a consequence we derive a classification of toric vector bundles
phrased in terms of tropicalized linear spaces. We use these tools to give a
characterization of the Mori dream space property for a projectivized toric
vector bundle.
