In this paper, the consensus problem of multiple nonholonomic mobile robots is addressed under event-triggered communication. It is assumed that the position of each robot is subject to generic time-varying constraints, and that the communication graph contains a directed spanning tree. A novel distributed algorithm that employs coordinate transformation with transverse functions and dynamic systems to generate reference signals is presented. Rigorous analysis, based on appropriately constructed Lyapunov functions, demonstrates that all signals in the closed-loop system remain bounded and that the consensus error can be regulated within a neighborhood of the origin. Moreover, we prove that the time-varying position constraints are consistently satisfied. The existence of a positive minimum inter-event time is also established for preventing Zeno behavior. The efficacy of the proposed approach is validated through comprehensive simulation studies involving five mobile robots.