Non-existence of Cannon-Thurston maps for hierarchically hyperbolic groups

We prove that Cannon--Thurston maps do not exist for hyperbolic normal subgroups of hierarchically hyperbolic groups, both for Morse and hierarchically hyperbolic boundaries. This result includes a large class of normal subgroups of the mapping class group of a surface of genus $g\geq 2$. As a corollary, we also prove a non-existence result for subgroups of most free by cyclic groups. Moreover, we prove that the visual boundary of a CAT(0) group with isolated flats is homeomorphic to the relatively hierarchically hyperbolic boundary of the space it acts on, thus recovering a previously known non-existence result for CAT(0) groups with isolated flats.