The problem of the upper semi-continuous dependence of global attractors of (nonautonomous) dynamical systems on parameters is well studied in the literature. However, the question of the continuous dependence is much less developed. The authors contribute to this question in the context of nonautonomous dynamical systems by proving a continuity theorem for global attractors under the assumption that the system is uniformly contracting with respect to the parameter. In addition, they show that infinite iterated function systems fit into the context of nonautonomous systems, which leads to generalization of the theorem of Barnsley concerning the continuous dependence of fractals on parameters.