  We develop a "weak Wa\.zewski principle" for discrete and continuous time
dynamical systems on metric spaces having a weaker topology to show that
attractors can be continued in a weak sense. After showing that the Wasserstein
space of a proper metric space is weakly proper we give a sufficient and
necessary condition such that a continuous map (or semiflow) induces a
continuous map (or semiflow) on the Wasserstein space. In particular, if these
conditions hold then the global attractor, viewed as invariant measures, can be
continued under Markov-type random perturbations which are sufficiently small
w.r.t. the Wasserstein distance, e.g. any small bounded Markov-type noise and
Gaussian noise with small variance will satisfy the assumption.
