Endowed with the Chabauty topology, the space of subgroups of any infinite countable group G is a closed subspace of the Cantor set, equipped with an action by homeomorphisms given by the $G$-conjugation. We are interested in the dynamics induced by this action on closed $G$-invariant subspaces. The largest closed subspace without isolated point is an example of such subspace called the perfect kernel of $G$. In an acylindrically hyperbolic context, Hull, Mynasyan and Osin demonstrated strong mixing properties (namely $\mu$-mixing for a suitable measure $\mu$ on $G$, a strengthening of high topological transitivity). We uncover a radically different situation in the case of non metabelian Baumslag-Solitar groups. For the decomposition of the perfect kernel introduced by Carderi, Gaboriau, Le Maоtre and Stalder, who proved high topological transitivity on each piece, we show that the conjugation is even $\mu$ -mixing in the case of unimodular Baumslag-Solitar groups. On the contrary, when the group is non unimodular, there exists a continuum of measures $\mu$ for which the action is $\mu$-mixing only on a single piece of the partition.
