Continuous time financial market models are often motivated as scaling limits of discrete time models. The objective of this paper is to establish such a connection for a robust framework. More specifically, we consider discrete time models that are parameterized by Markovian transition kernels, and a continuous time framework with drift and volatility uncertainty, again parameterized in a Markovian way. Our main result is a limit theory that establishes convergence of the uncertainty sets in the Hausdorff metric topology and weak convergence of the associated worst-case expectations. Furthermore, we discuss a structure preservation property of certain approximations. Namely, we establish the convergence of discrete to continuous time robust superhedging prices for some complete robust market models. As illustration of our main results, we use the idea of Kushner's Markov chain approximation method and provide a recursive algorithm for the computation of continuous time robust superhedging prices.