  We consider the case in which mixed graphs (with both directed and undirected
edges) are Cayley graphs of Abelian groups. In this case, some Moore bounds
were derived for the maximum number of vertices that such graphs can attain. We
first show these bounds can be improved if we know more details about the order
of some elements of the generating set. Based on these improvements, we present
some new families of mixed graphs. For every fixed value of the degree, these
families have an asymptotically large number of vertices as the diameter
increases. In some cases, the results obtained are shown to be optimal.
