  The Dunford property $(C)$ for composition operators on $H^p$-spaces
($1<p<\infty$), as well as for their adjoints, is completely characterized
within the class of those induced by linear fractional transformations of the
unit disc. As a consequence, it is shown that the Dunford property is stable in
such a class addressing a particular instance of a question posed by Laursen
and Neumann.
