Stability analysis for optimization problems with chance constraints concerns impact of variation of probability measure in the chance constraints on the optimal value and optimal solutions and research on the topic has been well documented in the literature of stochastic programming. In this paper, we extend such analysis to optimization problems with distributionally robust chance constraints where the true probability is unknown, but it is possible to construct an ambiguity set of distributions and the chance constraint is based on the most conservative selection of probability distribution from the ambiguity set. The stability analysis focuses on impact of the variation of the ambiguity set on the optimal value and optimal solutions. We start by looking into continuity of the robust probability function and followed with a detailed analysis of approximation of the function. Sufficient conditions have been derived for continuity of the optimal value and outer semicontinuity of optimal solution set. Case studies are carried out for ambiguity sets being constructed through moments and samples.