Massive outflows of neutral atomic hydrogen (HI) have been observed in absorption in a number of radio galaxies and are considered a signature of active galactic nucleus (AGN) feedback. These outflows on kiloparsec scales have not been investigated in great detail as they require high-angular-resolution observations to be spatially resolved. In some radio AGN, they are likely to be the result of the radio jets interacting with the interstellar medium. We have used the global very-long-baseline-interferometry (VLBI) array to map the HI outflow in a small sample of young and restarted radio galaxies that we previously observed with the Very Large Array (VLA) and the Westerbork Synthesis Radio Telescope (WSRT) at a lower resolution. Here we report on our findings for 4C 52.37 and 3C 293 and we discuss the sample including the previously published 4C 12.50 and 3C 236. For 4C 52.37, we present the first ever HI VLBI observations, which recovered the majority of the outflowing HI gas in the form of clouds toward the central 100pc of the AGN. The clouds are blueshifted by up to ~600km/s with respect to the systemic velocity. 3C 293 is largely resolved out in our VLBI observation, but toward the VLBI core we detect some outflowing HI gas blueshifted with respect to the systemic velocity by up to ~300km/s. We also find indications of outflowing gas toward the other parts of the western lobe suggesting that the HI outflow is extended. Overall, we find that the fraction of HI gas recovered by our VLBI observations varies significantly within our sample, ranging from complete (4C 12.50) to marginal (3C 293). However, in all cases we find evidence for a clumpy structure of both the outflowing and the quiescent gas, consistent with predictions from numerical simulations. All the outflows include at least a component of relatively compact clouds with masses in the range of 10^4^-10^5^M_{sun}_. The outflowing clouds are often already observed at a few tens of parsecs (in projection) from the core. We find indications that the HI outflow might have a diffuse component, especially in larger sources. Our results support the interpretation that we observe these AGNs at different stages in the evolution of the interaction between the jet and the interstellar medium and this is reflected in the properties of the outflowing gas as predicted by numerical simulations.