  In an earlier paper, we proved that given an asymptotically cylindrical
G_2-manifold M with a Calabi-Yau boundary X, the moduli space of coassociative
deformations of an asymptotically cylindrical coassociative 4-fold C in M with
a fixed special Lagrangian boundary L in X is a smooth manifold of dimension
dim(V_+), where V_+ is the positive subspace of the image of H^2_{cs}(C,R) in
H^2(C,R). In order to prove this we used the powerful tools of Fredholm Theory
for noncompact manifolds which was developed by Lockhart and McOwen, and
independently by Melrose.
  In this paper, we extend our result to the moving boundary case. Let
Upsilon:H^2(L,R)--> H^3_{cs}(C,R) be the natural projection, so that
ker(Upsilon) is a vector subspace of H^2(L,R). Let F be a small open
neighbourhood of 0 in ker(Upsilon) and L_s denote the special Lagrangian
submanifolds of X near L for some s in F and with phase i. Here we prove that
the moduli space of coassociative deformations of an asymptotically cylindrical
coassociative submanifold C asymptotic to L_s x (R,infinity), s in F, is a
smooth manifold of dimension equal to dim V_++dim(ker(Upsilon))=dim V_+
+b^2(L)-b^0(L)+b^3(C)-b^1(C)+b^0(C).
