  We consider three-partite pure states in the Hilbert space $\mathbb{C}^2
\otimes \mathbb{C}^m \otimes \mathbb{C}^n$ and investigate to which states a
given state can be locally transformed with a non-vanishing probability.
Whenever the initial and final states are elements of the same Hilbert space,
the problem can be solved via the characterization of the entanglement classes
which are determined via stochastic operations and classical communication
(SLOCC). In general, there are infinitely many SLOCC classes. However, when
considering transformations from higher- to lower-dimensional Hilbert spaces,
an additional hierarchy among the classes can be found. This hierarchy of SLOCC
classes coarse grains SLOCC classes which can be reached from a common resource
state of higher dimension. We first show that a generic set of states in
$\mathbb{C}^2 \otimes \mathbb{C}^m \otimes \mathbb{C}^n$ for $n=m$ is the union
of infinitely many SLOCC classes, which can be parameterized by $m-3$
parameters. However, for $n \neq m$ there exists a single SLOCC class which is
generic. Using this result, we then show that there is a full-measure set of
states in $\mathbb{C}^2 \otimes \mathbb{C}^m \otimes \mathbb{C}^n$ such that
any state within this set can be transformed locally to a full measure set of
states in any lower-dimensional Hilbert space. We also investigate resource
states, which can be transformed to any state (not excluding any zero-measure
set) in the smaller-dimensional Hilbert space. We explicitly derive a state in
$\mathbb{C}^2 \otimes \mathbb{C}^m \otimes \mathbb{C}^{2m-2}$ which is the
optimal common resource of all states in $\mathbb{C}^2 \otimes \mathbb{C}^m
\otimes \mathbb{C}^m$. We also show that for any $n < 2m$ it is impossible to
reach all states in $\mathbb{C}^2 \otimes \mathbb{C}^m \otimes
\mathbb{C}^{\tilde{n}}$ whenever $\tilde{n}>m$.
