We investigate the formation of infant globular cluster (GC) candidates in high-resolution cosmological simulations from the First Billion Years project. By analysing the evolution of the systems in the energy and angular momentum plane, we identify the redshift at which the infant GCs first became gravitationally bound, and we find evidence of radial infall of their gaseous and stellar components. The collapse appears to be driven by internal self-gravity; however, the initial trigger is sourced from the external environment. The phase space behaviour of the infant GCs also allows us to identify some characteristic groupings of objects. Such a classification based on internal properties appears to be reflected in the formation environment: GC candidates that belong to the same class are found in host galaxies of similar morphology, with the majority of the infant GCs located in clumpy, irregular proto-galaxies. Finally, through the inspection of two GC candidates that contain only stars by z = 6, we find that supernova feedback is the main physical mechanism behind their dearth of gas and that the systems subsequently respond with an approximately adiabatic expansion. Such infant GC candidates already resemble the GCs we currently observe in the local Universe.