Model specification: Poisson random-Clifford circuit

Let $q=2^k$. Each site has $k$ qubits. Every edge of a finite graph has an independent rate-$\lambda$ Poisson clock. At each clock event apply an independent uniformly chosen two-site Clifford gate. Start with a nonidentity Pauli operator at $0$ and identity elsewhere. Let $S_q(t)$ be the support of its forward conjugate by the circuit.