  We establish the principle of specialization of integral dependence for
submodules of finite colength of free modules, as part of the general
algebraic-geometric theory of the Buchsbaum--Rim multiplicity. Then we apply
the principle to the study of equisingularity of ICIS germs, obtaining results
for such equisingularity conditions as Whitney's Condition A, Thom's Condition
A_f, and Henry, Merle and Sabbah's Condition W_f. Notably, we describe these
conditions for analytic families in terms of various numerical invariants,
which, for the most part, depend only on the members of a family, not on its
total space.
