  The main purpose of this paper is to extend results, which have been obtained
previously to describe the classical scattering of solitons with integrable
defects of type I, to include the much larger and intricate collection of
finite-gap solutions defined in terms of generalised theta functions. In this
context, it is generally not feasible to adopt a direct approach, via ansatze
for the fields to either side of the defect tuned to satisfy the defect sewing
conditions. Rather, essential use is made of the fact that the defect sewing
conditions themselves are intimately related to Backlund transformations in
order to set up a strategy to enable the calculation of the field on one side
by suitably transforming the field on the other side. The method is implemented
using Darboux transformations and illustrated in detail for the sine-Gordon and
KdV models. An exception, treatable by both methods, indirect and direct, is
provided by the genus 1 solutions. These can be expressed in terms of Jacobi
elliptic functions, which satisfy a number of useful identities of relevance to
this problem. There are new features to the solutions obtained in the
finite-gap context but, in all cases, if a (multi)soliton limit is taken within
the finite-gap solutions previously known results are recovered.
