The effect of disorder on the Berezinski-Kosterlitz-Thouless transition is still under debate, as numerical data is limited due to the exponentially divergent correlation length. We explore this question from the dynamical angle, by quenching a disordered system across the transition temperature, asking how the time dependence of the correlation length is modified by disorder. Such quenches can provide an experimentally accessible way to measure disorder-modified critical behavior.