Context. The assembly history experienced by the Milky Way is currently being unveiled thanks to the data provided by the Gaia mission. It is likely that the globular cluster system of our Galaxy has followed a similarly intricate formation path. Aims. To constrain this formation path, we explore the link between the globular clusters and the known merging events that the Milky Way has experienced. Methods. To this end, we combined the kinematic information provided by Gaia for almost all Galactic clusters, with the largest sample of cluster ages available after carefully correcting for systematic errors. To identify clusters with a common origin we analysed their dynamical properties, particularly in the space of integrals of motion. Results. We find that about 40% of the clusters likely formed in situ. A similarly large fraction, 35%, appear to be possibly associated to known merger events, in particular to Gaia-Enceladus (19%), the Sagittarius dwarf galaxy (5%), the progenitor of the Helmi streams (6%), and to the Sequoia galaxy (5%), although some uncertainty remains due to the degree of overlap in their dynamical characteristics. Of the remaining clusters, 16% are tentatively associated to a group with high binding energy, while the rest are all on loosely bound orbits and likely have a more heterogeneous origin. The resulting age–metallicity relations are remarkably tight and differ in their detailed properties depending on the progenitor, providing further confidence on the associations made. Conclusions. We provide a table listing the likely associations. Improved kinematic data by future Gaia data releases and especially a larger, systematic error-free sample of cluster ages would help to further solidify our conclusions.