  We introduce a notion of "local stability in permutations" for finitely
generated groups. If a group is sofic and locally stable in our sense, then it
is also locally embeddable into finite groups (LEF). Our notion is weaker than
the "permutation stability" introduced by Glebsky-Rivera and
Arzhantseva-Paunescu, which allows one to upgrade soficity to residual
finiteness. We prove a necessary and sufficient condition for an amenable group
to be locally permutation stable, in terms of invariant random subgroups
(IRSs), inspired by a similar criterion for permutation stability due to
Becker, Lubotzky and Thom. We apply our criterion to prove that derived
subgroups of topological full groups of Cantor minimal subshifts are locally
stable, using Zheng's classification of IRSs for these groups. This last result
provides continuum-many groups which are locally stable, but not stable.
