Let $μ$ = ($μ$t)t$\in$R be a 1-parameter family of probability measures on R. In [11] we introduced its ``Markov-quantile''process: a process X= (Xt)t$\in$R that resembles as much as possible the quantile process attached to $μ$, among the Markov processesattached to $μ$, i.e. whose family of marginal laws is $μ$.In this article we look at the case where $μ$ is absolutely continuous in the Wasserstein space P2(R). Then X is solution of adynamical transport problem with marginals ($μ$t)t. It provides a Markov minimal Lagrangian probabilistic representative of $μ$, whichis moreover unique among the processes obtained as certain types of limits: limits for the finite dimensional topology of quantileprocesses where the past is made independent of the future conditionally on the present at finitely many times, or limits of processeslinearly interpolating $μ$.This raises new questions about ways to obtain Markov Lagrangian representatives, and to seek uniqueness properties in thisframework.