Given a finite transitive graph and $n$ particles labeled with numbers ${1,2,3,\dots,n}$, we place these particles on the set of vertices at random. Then, we let the particles evolve as a system of coalescing random walks: each particle performs a continuous-time simple random walk (SRW) and whenever two particles meet, they merge into one particle which continues to perform a SRW. At each time t, consider the partition $P_t$ of ${1,2,3,\dots,n}$ induced by the equivalence relation: $i\sim j$ when particles $i$ and $j$ occupy the same vertex at time $t$. We show that the Kingman $n$-coalescent model emerges as a scaling limit for $(P_t)$, as $n$ is fixed and the size of the graph goes to infinity.