  The study of state transformations by spatially separated parties with local
operations assisted by classical communication (LOCC) plays a crucial role in
entanglement theory and its applications in quantum information processing.
Transformations of this type among pure bipartite states were characterized
long ago and have a revealing theoretical structure. However, it turns out that
generic fully entangled pure multipartite states cannot be obtained from nor
transformed to any inequivalent fully entangled state under LOCC. States with
this property are referred to as isolated. Nevertheless, the above result does
not forbid the existence of particular SLOCC classes that are free of
isolation, and therefore, display a rich structure regarding LOCC
convertibility. In fact, it is known that the celebrated $n$-qubit GHZ and W
states give particular examples of such classes and in this work, we
investigate this question in general. One of our main results is to show that
the SLOCC class of the 3-qutrit totally antisymmetric state is isolation-free
as well. Actually, all states in this class can be converted to inequivalent
states by LOCC protocols with just one round of classical communication (as in
the GHZ and W cases). Thus, we consider next whether there are other classes
with this property and we find a large set of negative answers. Indeed, we
prove weak isolation (i.e., states that cannot be obtained with finite-round
LOCC nor transformed by one-round LOCC) for very general classes, including all
SLOCC families with compact stabilizers and many with non-compact stabilizers,
such as the classes corresponding to the $n$-qunit totally antisymmetric states
for $n\geq4$. Finally, given the pleasant feature found in the family
corresponding to the 3-qutrit totally antisymmetric state, we explore in more
detail the structure induced by LOCC and the entanglement properties within
this class.
