Dominance links were introduced in grammars to model long distance scrambling phenomena, motivating the definition of multiset-valued linear indexed grammars (MLIGs) by Rambow (ACL 1994), and inspiring quite a few recent formalisms. It turns out that MLIGs have since been rediscovered and reused in a variety of contexts, and that the complexity of their emptiness problem has become the key to several open questions in computer science. We survey complexity results and open issues on MLIGs and related formalisms, and provide new complexity bounds for some linguistically motivated restrictions.