  We introduce several different notions of disjoint distributional chaos for
sequences of multivalued linear operators in Fr\'echet spaces. Any of these
notions seems to be new and not considered elsewhere even for linear continuous
operators in Banach spaces. We focus special attention to the analysis of some
specific classes of linear continuous operators having a certain disjoint
distributionally chaotic behaviour, providing also a great number of
illustrative examples and applications of our abstract theoretical results.
