  This is the third and last in a series of papers working towards the
construction of non-trivial Cayley fibrations using gluing methods. In this
paper we will show two stability results for Cayley fibrations with certains
types of conical singularities (in particular Morse type singularities present
in holomorphic fibrations of Calabi--Yau fourfolds). The first is a stability
result for weak fibrations, which has minimal assumptions. Then we show
stability of Cayley fibrations in the usual sense. This requires stronger
geometric assumptions on the Cayley cone and the initial fibration. As an
application we construct examples of Cayley fibrations on twisted connected sum
$G_2$ manifolds times a circle. In particular we also obtain examples of
coassociative fibrations of twisted connected sum $G_2$ manifolds, completing
the longstanding programme by Kovalev.
