We will present graph-theoretic results appropriate for the analysis of a variety of consensus problems cast in dynamically changing environments. The concepts of rooted, strongly rooted, and neighbor-shared graphs will be defined, and conditions will be derived for compositions of sequences of directed graphs to be of these types. As an illustration of the use of the concepts covered, graph theoretic conditions will be derived which address the convergence question for the leaderless version of the widely studied flocking problem. We will also explain how to use these graph-theoretic constructions to address modified versions of the flocking problem in which there are measurement delays, asynchronous events, or a group leader. In all three cases the conditions under which consensus is achieved prove to be almost the same as the conditions under which consensus is achieved in the synchronous, delay-free, leaderless case.