In dynamical percolation, the status of every bond is refreshed according to an independent Poisson clock. For graphs which do not percolate at criticality, the dynamical sensitivity of this property was analyzed extensively in the last decade. Here we focus on graphs which percolate at criticality, and investigate the dynamical sensitivity of the infinite cluster. We first give two examples of bounded degree graphs, one which percolates for all times at criticality and one which has exceptional times of nonpercolation. We then make a nearly complete analysis of this question for spherically symmetric trees with spherically symmetric edge probabilities bounded away from 0 and 1. One interesting regime occurs when the expected number of vertices at the nth level that connect to the root at a fixed time is of order n(log n)α. R. Lyons (1990) showed that at a fixed time, there is an infinite cluster a.s. if and only if α>1. We prove that the probability that there is an infinite cluster at all times is 1 if α>2, while this probability is 0 if 1<α≤2. Within the regime where a.s. there is an infinite cluster at all times, there is yet another type of “phase transition” in the behavior of the process: if the expected number of vertices at the nth level connecting to the root at a fixed time is of order nθ with θ>2, then the number of connected components of the set of times in [0, 1] at which the root does not percolate is finite a.s., while if 1<θ<2, then the number of such components is infinite with positive probability.