We introduce quivers of valuated matroids and study their tropical parameter spaces. We define quiver Dressians, which parametrize containment of tropical linear spaces after tropical matrix multiplication, and show that tropicalizations of quiver Grassmannians parametrize the realizable analogue. We further introduce affine morphisms of valuated matroids and show compatibility with weakly monomial quiver representations. Finally, we show that starting in ambient dimension 2, quiver Dressians can have nonrealizable points.