  We define and study the natural multigraded extension of the relative
multiplicities introduced by Simis, Ulrich and Vasconcelos. We call these new
invariants relative mixed multiplicities. We show that they have a stable value
equal to the mixed Buchsbaum-Rim multiplicity of Kleiman and Thorup.
Furthermore, we prove that integral dependence and birationality can be
detected via the vanishing of relative mixed multiplicities.
