  We prove that each local Lie $n$-derivation is a Lie $n$-derivation under
mild assumptions on the unital algebras with a nontrivial idempotent. As
applications, we obtain descriptions of local Lie $n$-derivations on
generalized matrix algebras, triangular algebras, nest algebras, von Neumann
algebras, and the algebras of locally measurable operators affiliated with a
von Neumann algebra.
