  We investigate the effects of smooth inhomogeneities at first-order quantum
transitions (FOQT), such as those arising from the presence of a
space-dependent external field, which smooths out the typical discontinuities
of the low-energy properties. We argue that scaling phenomena develop at the
transition region where the external field takes the value corresponding to the
FOQT of the homogenous system. We present numerical evidence of such scaling
phenomena at the FOQTs of quantum Ising chains, driven by a parallel magnetic
field when the system is in the ferromagnetic phase, and at the FOQT of the
q-state Potts chain for q>4, driven by an even temperature-like parameter
giving rise to a discontinuity of the ground-state energy density.
