We discuss a Monte Carlo Markov Chain (MCMC) procedure for the random sampling of some one-dimensional lattice paths with constraints, for various constraints. We show that an approach inspired by optimal transport allows us to bound efficiently the mixing time of the associated Markov chain. The algorithm is robust and easy to implement, and samples an "almost" uniform path of length $n$ in $n^{3+\eps}$ steps. This bound makes use of a certain contraction property of the Markov chain, and is also used to derive a bound for the running time of Propp-Wilson's CFTP algorithm.