We present a framework for the design of time- and event-triggered communication schemes for a broad class of multi-agent systems (MAS). The framework is general in the sense that we consider a class of heterogeneous nonlinear MAS with directed communication graphs subject to disturbances and the inevitable imperfections induced by packet-based networked communication, such as time-varying transmission delays and intervals. Under suitable conditions, the designed distributed time- or event-triggered communication schemes lead to a MAS that is dissipative with respect to a desired supply rate, and thus can be used to study various properties such as Lp-gain stability, input-to-state stability and passivity. Strictly positive lower bounds on the inter-event times are ensured and robustness to non-uniform time-varying delays in terms of maximum allowable delays is guaranteed. In addition, the framework allows for the consideration of destination protocols which determine, at each transmission instant, to which (groups of) agent(s) a local output measurement is being transmitted. These destination protocols can be used to include packet loss and denial-of-service, as shown in the companion paper (Part II).