  Work of Gonz\'alez, Hering, Payne, and S\"uss shows that it is possible to
find both examples and non-examples of Mori dream spaces among projectivized
toric vector bundles. This result, and the combinatorial nature of the data of
projectivized toric vector bundles make them an ideal test class for the
question: what makes a variety a Mori dream space? In the present paper we
consider this question with respect to natural algebraic operations on vector
bundles.
  Suppose $\mathcal{E}$ is a toric vector bundle such that the projectivization
$\mathbb{P}\mathcal{E}$ is a Mori dream space, then when are the direct sum
bundles $\mathbb{P}(\mathcal{E} \oplus \mathcal{E})$, $\mathbb{P}(\mathcal{E}
\oplus \mathcal{E} \oplus \mathcal{E})\ldots$ also Mori dream spaces? We give
an answer to this question utilizing a relationship with the associated full
flag bundle $\mathcal{FL}(\mathcal{E})$. We describe several classes of
examples, and we compute a presentation for the Cox ring of the full flag
bundle for the tangent bundle of projective space.
