We present a detailed comparison of the Milky Way (MW) globular cluster (GC) kinematics with the 25 Milky Way-mass cosmological simulations from the E-MOSAICS project. While the MW falls within the kinematic distribution of GCs spanned by the simulations, the relative kinematics of its metal-rich ($[\rm {Fe}/\rm {H}]>-1.2$) versus metal-poor ($[\rm {Fe}/\rm {H}]<-1.2$), and inner (r < 8 kpc) versus outer (r > 8 kpc) populations are atypical for its mass. To understand the origins of these features, we perform a comprehensive statistical analysis of the simulations, and find 18 correlations describing the assembly of L* galaxies and their dark matter haloes based on their GC population kinematics. The correlations arise because the orbital distributions of accreted and in-situ GCs depend on the masses and accretion redshifts of accreted satellites, driven by the combined effects of dynamical fraction, tidal stripping, and dynamical heating. Because the kinematics of in-situ/accreted GCs are broadly traced by the metal-rich/metal-poor and inner/outer populations, the observed GC kinematics are a sensitive probe of galaxy assembly. We predict that relative to the population of L* galaxies, the MW assembled its dark matter and stellar mass rapidly through a combination of in-situ star formation, more than a dozen low-mass mergers, and 1.4 ± 1.2 early (z = 3.1 ± 1.3) major mergers. The rapid assembly period ended early, limiting the fraction of accreted stars. We conclude by providing detailed quantitative predictions for the assembly history of the MW.