The most recently formed young stellar objects (YSOs) in active star-forming regions are excellent tracers of their parent cloud motion. Their positions and dynamics provide insights into cluster formation and constrain kinematic decoupling timescales between stars and gas. However, because of their strong extinction and young age, embedded YSOs are mainly visible at infrared wavelengths, and are thus absent from astrometric surveys such as Gaia. We measured the proper motions of 6769 sources toward the NGC 2024 cluster in the Flame Nebula (d~420pc) using multi-epoch near-infrared observations from three ESO public surveys: VISIONS, VHS, and the VISTA/VIRCAM science verification program. Cross-validation of our results with Gaia using optically visible stars shows excellent agreement, with uncertainties on the same order of magnitude. For 362 YSO candidates identified from the literature, we derived proper motions on the order of <5mas/yr with mean measurement uncertainties of 0.22mas/yr, or 0.44km/s. This is the first homogeneous proper motion measurement of this quality for more than half of these stars. For Class I and flat-spectrum sources, our results provide a >13-fold increase in available proper motion measurements. We analyzed the positional and kinematic differences between YSO classes and confirmed a previously reported inside-out age segregation from younger to older stars, likely driven by an outward movement of older stars. No evidence of prolonged hierarchical assembly was found. Instead, the results support a rapid (<1Myr) cluster collapse into a centrally concentrated system. This scenario also accounts for the observed slightly higher 1D velocity dispersion of Class I sources relative to Class flat objects. YSO radial velocities generally align with the gas velocities measured from the molecular transitions of ^12^CO(3-2), HNC(1-0), HCN(1-0), and show a weaker correlation with N_2_H+(1-0). Some Class II and III objects already appear to be decoupling.