Continuity of Furstenberg entropy on Bowen Poisson bundles

We provide a sufficient criterion for continuity of the Furstenberg entropy of Bowen Poisson bundles, jointly in the random walk measure and the underlying invariant random subgroup. The criterion applies in particular for tree-like Poisson bundles of a free group and for intersectional invariant random subgroups on lamplighter groups. It also provides continuity of the boundary entropy spectrum of powers of prime measured groups.