Timely state updates are essential for coordination and monitoring in multi-agent systems, such as teams of mobile robots exchanging state information. When agents share a wireless channel, collisions and slow convergence of distributed access schemes can degrade information freshness over finite operating periods. This paper proposes age-threshold frame-based random access (AFRA), a distributed medium access control protocol for minimizing age of information (AoI). Once its age reaches a threshold, an agent starts a local frame with a mandatory transmission and, if unsuccessful, makes one further attempt at a uniformly selected remaining slot. Each attempt carries a freshly generated update. AFRA requires only local state and acknowledgment feedback, with shared slot boundaries but independently timed local frames. We prove almost-sure convergence from any legal initial state to collision-free periodic operation when the frame length $L$ and age threshold $T$ satisfy $L\ge T>N\ge2$, where $N$ is the number of agents. We also derive closed-form expressions for steady-state throughput, average AoI, and average peak AoI. Simulations demonstrate that AFRA combines rapid convergence at small age thresholds with lower window-averaged AoI than the existing age-aware random access schemes evaluated in this study.

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