  We consider the class of (possibly killed) spectrally positive L\'evy process
that have been time-changed by the inverse of an integral functional. Within
this class we characterize the family of those processes which satisfy the
following property: as functions of point of issue, the Laplace transforms of
their first-passage times downwards are completely monotone. A wide (dense, in
a sense) subfamily of this family admits closed form expressions for said
Laplace transforms.
