  Stacky Lie groupoids are generalizations of Lie groupoids in which the "space
of arrows" of the groupoid is a differentiable stack. In this paper, we
consider actions of stacky Lie groupoids on differentiable stacks and their
associated quotients. We provide a characterization of principal actions of
stacky Lie groupoids, i.e., actions whose quotients are again differentiable
stacks in such a way that the projection onto the quotient is a principal
bundle. As an application, we extend the notion of Morita equivalence of Lie
groupoids to the realm of stacky Lie groupoids, providing examples that
naturally arise from non-integrable Lie algebroids.
