A well-known consequence of the Prékopa-Leindler inequality is the preservation of log-concavity by the heat semigroup. Unfortunately, this property does not hold for more general semigroups. In this talk, leveraging the probabilistic notion of reflection coupling, I will present a slightly weaker notion of log-concavity that can be propagated along generalised heat semigroups. As a consequence, log-semiconcavity properties for the ground state of Schrödinger operators for non-convex potentials, propagation of functional inequalities along generalised heat flows and log-Hessian estimates for fundamental solutions can be obtained in non log-concave settings. This is a joint work with Giovanni Conforti and Katharina Eichinger.