We consider the stochastic evolution of three variants of the RSK algorithm, giving both analytic descriptions and probabilistic interpretations. Symmetric functions play a key role, and the probabilistic interpretations are obtained by elementary Doob-Hunt theory. In each case, the evolution of the shape of the tableau obtained via the RSK algorithm can be interpreted as a conditioned random walk. This is intuitively appealing, and can be used for example to obtain certain relationships between orthogonal polynomial ensembles. In a certain scaling limit, there is a continous version of the RSK algorithm which inherits much of the structure exhibited in the discrete settings. Intertwining relationships between conditioned and unconditioned random walks are also given. In the continuous limit, these are related to the Harish-Chandra/Itzyksen-Zuber integral.