  We extend the construction of higher mutation as introduced in
Pascaleff-Tonkonog to local higher mutation, which is applicable to a larger
class of monotone Lagrangians. In two-dimensional Lagrangians, local higher
mutation is the same as performing a Lagrangian anti-surgery in the sense of
Haug followed by a Lagrangian surgery. We prove that up to a change of local
systems, the Lagrangian intersection Floer cohomology of a pair of Lagrangians
is invariant under local mutation. This result generalizes the wall-crossing
formula in Pascaleff-Tonkonog. For two-dimensional Lagrangians, this result
agrees with the invariance result in Palmer-Woodward.
