  This paper is a continuation of an earlier paper of the authors on the probem
of specializations of modules. The aim here is to define specialisations of
finitely generated modules over local rings of the form k(u)[X]_P, where u is a
family of indeterminates over a field k. The definition depends on the
associated primes of the specialisation of the prime ideal P by the
substitution of u to elements of k. As in the non-local case, the introduced
specializations preserve basic properties and operations on modules.In
particular, annihilators of local cohomology modules commute with
specializations. As a consequence, Cohen-Macaulayness and Buchsbaumness are
preserved by almost al substitutions. The paper also studies the
non-Cohen-Macaulay locus and the singular locus of specializations of factor
rings of k(u)[X]_P.
