The production yields of the X(3872) particle in heavy-ion collisions have been suggested to be sensitive to its internal structure. While the thermal equilibrium limit of the X(3872) abundance only depends on its mass, its inelastic reaction rates in a hot medium are expected to be different depending on whether it is a loosely bound hadronic molecule or a compact tetraquark dominated by colored diquark-antidiquark configurations. Here we implement these scenarios into a kinetic rate equation as previously used in applications to charmonium and bottomonium observables. While the equilibrium limit, controlled by the charm-quark content in the fireball, is universal for both scenarios (and decreases with temperature), the molecular scenario is characterized by a much larger width in the hadronic medium than for the tetraquark. Consequently, the average production time of the molecular X(3872) is significantly later than for the tetraquark, leading to to a factor of $\sim$2 smaller production of the former compared to the latter. This outcome is qualitatively different from most coalescence model calculations to date. In addition, we calculate pertinent transverse-momentum spectra, which turn out to be harder for the molecular scenario due to the larger blue-shift caused by the collective flow in the fireball at later times.