  Let X/T be a one parameter family of canonical 3-folds and let D be a Weil
divisor on it flat over T. We study the problem of when the D_t-minimal models
of X_t form a family and we obtain conditions for this to happen. As an
application of this we classify terminal divisorial contractions Y-->X that
contract an irreducible divisor E onto a smooth curve C in the case when the
general hyperplane section S of X through C is a D5 DuVal singularity.
