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We prove that, in stable families of endomorphisms of$\mathbb{P}^k(\mathbb{C})$, all invariant measures whose measure-theoreticentropy is strictly larger than $(k-1)\log d$ at a given parameter can befollowed holomorphically with the parameter in all the parameter space. As aconsequence, almost all points (with respect to any such measure at anyparameter) in the Julia set can be followed holomorphically withoutintersections. This generalizes previous results by Berteloot, Dupont, and thefirst author for the measure of maximal entropy, and provides a parallel inthis setting to the probabilistic stability of H\'enon maps byBerger-Dujardin-Lyubich. Our proof relies both on techniques from the theory ofstability/bifurcation in any dimension and on an explicit lower bound for theLyapunov exponents for an ergodic measure in terms of its measure-theoreticentropy, due to de Th\'elin and Dupont. A local version of our result holdsalso for all measures supported on the Julia set with just strictly positiveLyapunov exponents and not charging the post-critical set. Analogous resultshold in families of polynomial-like maps of large topological degree. In thiscase, as part of our proof, we also give a sufficient condition for thepositivity of the Lyapunov exponents of an ergodic measure for apolynomial-like map in any dimension in term of its measure-theoretic entropy,generalizing to this setting the analogous result by de Th\'elin and Dupontvalid on $\mathbb{P}^k(\mathbb{C})$.

Strong probabilistic stability in holomorphic families of endomorphisms of $\mathbb{P}^k(\mathbb{C})$ and polynomial-like maps