A well-known consequence of the Pr{é}kopa-Leindler inequality is the preservation of logconcavity by the heat semigroup. Unfortunately, this property does not hold for more general semigroups. In this paper, we exhibit a slightly weaker notion of log-concavity that can be propagated along generalised heat semigroups. As a consequence, we obtain logsemiconcavity properties for the ground state of Schr{ö}dinger operators for non-convex potentials, as well as propagation of functional inequalities along generalised heat flows. We then investigate the preservation of weak log-concavity by conditioning and marginalisation, following the seminal works of Brascamp and Lieb. To our knowledge, our results are the first of this type in non log-concave settings. We eventually study generation of log-concavity by parabolic regularisation and prove novel two-sided log-Hessian estimates for the fundamental solution of parabolic equations with unbounded coefficients, which can be made uniform in time. These properties are obtained as a consequence of new propagation of weak convexity results for quadratic Hamilton-Jacobi-Bellman (HJB) equations. The proofs rely on a stochastic control interpretation combined with a second order analysis of reflection coupling along HJB characteristics.