We introduce a new metric for collections of aged paths and a robust set of conditions implying compactness for set of collections of aged paths in the topology corresponding to this metric. We show that the distribution of stable webs (1<α≤2) made up of collections of stable paths is tight in this topology. We then show convergence to stable webs for coalescing systems of random walks(suitably normalized). We obtain some path results in the Brownian case.