  We consider random switching between finitely many vector fields leaving
positively invariant a compact set. Recently, Li, Liu and Cui showed that if
one the vector fields has a globally asymptotically stable (G.A.S.) equilibrium
from which one can reach a point satisfying a weak H\"ormander-bracket
condition, then the process converges in total variation to a unique invariant
probability measure. In this note, adapting the proof of Li, Liu and Cui and
using results of Bena\"im, Le Borgne, Malrieu and Zitt, the assumption of a
G.A.S. equilibrium is weakened to the existence of an accessible point at which
a barycentric combination of the vector fields vanishes. Some examples are
given which demonstrate the usefulness of this condition.
