This paper presents the design and analysis result for asynchronous rendezvous control of multiagent systems with continuous-time dynamics and intermittent interactions. The protocol-designing strategies only impose weak restrictions on anticipated-way-point sets (from which the way-points are selected) and path-planning of each agent and can be applied to the networks of arbitrary dimensional subsystems. Explicitly, the anticipated-way-point sets are in a polytope-like form and the path between any two consecutive way-points is required to be included within the minimum convex region covering the two associated anticipated-way-point sets. Under the assumption of directed and switching interaction topology and the assumption of intermittent and asynchronous interactions with time-varying delays, we perform the set-valued consensus analysis on the evolution of anticipated-way-point sets with respect to update times and provide mild sufficient conditions for the solvability of the asynchronous rendezvous problem. The proof techniques rely much on graph theory and nonnegative matrix theory. The obtained result extends greatly the existing work in the literature and several examples demonstrate its broad potential applications. Particularly, additional distributed control rules, different from the circumcenter algorithm, are devised for network connectivity maintenance.