  We investigate the process of phase conversion in a thermally-driven {\it
weakly} first-order quark-hadron transition. This scenario is physically
appealing even if the nature of this transition in equilibrium proves to be a
smooth crossover for vanishing baryonic chemical potential. We construct an
effective potential by combining the equation of state obtained within Lattice
QCD for the partonic sector with that of a gas of resonances in the hadronic
phase, and present numerical results on bubble profiles, nucleation rates and
time evolution, including the effects from reheating on the dynamics for
different expansion scenarios. Our findings confirm the standard picture of a
cosmological first-order transition, in which the process of phase conversion
is entirely dominated by nucleation, also in the case of a weakly first-order
transition. On the other hand, we show that, even for expansion rates much
lower than those expected in high-energy heavy ion collisions, nucleation is
very unlikely, indicating that the main mechanism of phase conversion is
spinodal decomposition. Our results are compared to those obtained for a
strongly first-order transition, as the one provided by the MIT bag model.
