Let G be a graded graph with levels V₀, V₁,.. Fix m and choose a vertex v in Vn where n ≥ m. Consider the uniform measure on the paths from V₀ to v. Each such path has a unique vertex at the level V_m, so a measure νvm on V_m is induced. It is natural to expect that these measures have a limit as the vertex v goes to infinity in some “regular” way. We prove this (and compute the limit) for the Young and Schur graphs, for which regularity is understood as follows: the fraction of boxes contained in the first row and the first column goes to 0. For the Young graph, this was essentially proved by Vershik and Kerov in 1981; our proof is more straightforward and elementary.