Knothe-Rosenblatt maps provide an explicit construction to map between two probability measures and have shown to be useful in different realms of mathematics and statistics, from proving functional inequalities to designing methodology for sampling conditional distributions. It was shown in previous work that this map is the limit of a sequence of optimal transport maps with a weighted quadratic cost. We show that one can also obtain the Knothe-Rosenblatt map via a limit of maps that solve a weighted free target version of optimal transport. We extend this procedure to the construction of dynamic optimal transport with triangular velocity fields. This is ongoing work together with Ricardo Baptista, Minh Nguyen and Benjamin Zhang.