  We prove that, within any holomorphic family of endomorphisms of $\mathbb
P^k(\mathbb C)$ in any dimension $k \geq 1$ and algebraic degree $d \geq 2$,
the measurable holomorphic motion associated to dynamical stability in the
sense of Berteloot-Bianchi-Dupont preserves the class of equilibrium states
associated with weight functions $\psi$ satisfying $\sup\psi - \inf \psi < \log
d$.
