The strong stationary times introduced by Aldous and Diaconis (1986) provide a probabilistic approach to quantitative convergence to equilibrium. They are often obtained as the absorption times of intertwining dual processes, following a method due to Diaconis and Fill (1990). We will see how to deduce explicit constructions from certain random mappings, related to the coupling-from-the-past algorithm of Propp and Wilson (1996) and to the evolving sets of Morris and Peres (2005). This approach can be adapted, via the coalescing stochastic flows of Le Jan and Raimond (2006) associated to Tanaka's equation, to recover Pitman's theorem (1975) on the intertwining relation between the Brownian motion and the Bessel-3 process. Nevertheless the theory of coalescing stochastic flows is not sufficiently developed to go further in this direction and the talk will end with an alternative approach proposed in collaboration with Arnaudon and Coulibaly-Pasquier.