  We show the asymptotic relationship between the limit of the normalized
length function of a multi-$p-$family of ideals and that of its shifted family
under linear growth conditions in a local domain of characteristic $p$.
Examples of multi-$p-$families of ideals including products of Frobenius powers
of different ideals. We apply our results to obtain a generalized version of a
formula due to Wantanabe-Yoshida for certain $p-$families using results from
Verma, and to provide an instance of the existence of a mixed multiplicity
version of multi-$p$-families of ideals.
