Starting from a random environment of either a Poisson point process in the plane or grid of geometric random variables on the lattice, we use a variant of the Robinson-Schensted-Knuth (RSK) correspondence to create a random process of Young diagrams which are evolving in time and are closely related to the last passage percolation problem in these random environments. These processes can be equivalently thought of as a pair of "decorated" Young tableaux or as non-intersecting line ensembles. By simple properties of these particular random environments, one can show that the resulting processes have simple descriptions in terms of Schur symmetric functions. This makes these objects amenable to precise analysis and reveals the asymptotic behavior of the last passage percolation problem in these special environments.