We have derived the mean proper motions and space velocities of 154 Galactic globular clusters and the velocity dispersion profiles of 141 globular clusters based on a combination of Gaia DR2 proper motions with ground-based line-of-sight velocities. Combining the velocity dispersion profiles derived here with new measurements of the internal mass functions allows us to model the internal kinematics of 144 clusters, more than 90 per cent of the currently known Galactic globular cluster population. We also derive the initial cluster masses by calculating the cluster orbits backwards in time applying suitable recipes to account for mass-loss and dynamical friction. We find a correlation between the stellar mass function of a globular cluster and the amount of mass lost from the cluster, pointing to dynamical evolution as one of the mechanisms shaping the mass function of stars in clusters. The mass functions also show strong evidence that globular clusters started with a bottom-light initial mass function. Our simulations show that the currently surviving globular cluster population has lost about 80 per cent of its mass since the time of formation. If globular clusters started from a lognormal mass function, we estimate that the Milky Way contained about 500 globular clusters initially, with a combined mass of about 2.5x10^8^M_{sun}_. For a power-law initial mass function, the initial mass in globular clusters could have been a factor of three higher.