Young diagrams label irreducible representations of symmetric and unitary groups. Also, this is a fundamental concept in combinatorics and symmetric function theory. There are lots of beautiful formulas and algorithms related to Young diagrams and tableaux. It would be interesting to extend some of these constructions to continuous diagrams, i.e. uniform limits of (scaled) Young diagrams considered as piecewise linear functions. One can expect a connection with problems in function theory and analysis. The paper contains an example of such situation. It is shown that the correspondence between a Young diagram and its transition distribution in the Plancherel growth process extends by continuity to the bijection between the space of certain Lipschitz functions (continuous diagrams) and that of one-dimensional probability distributions. The latter bijection was known in a number of distinct setups, Markov moment problem and distributions of Dirichlet random integrals in particular. The Hook Walk Algorithm by Greene, Nijenhuis and Wilf gives rise to an entirely new description of the above correspondence via the interval shrinkage process introduced in the paper.