We investigate the internal kinematics of the young star-forming region NGC 346 in the Small Magellanic Cloud (SMC). We used two epochs of deep F555W and F814W Hubble Space Telescope Advanced Camera for Surveys observations with an 11 yr baseline to determine proper motions and study the kinematics of different populations, as identified by their color-magnitude diagram and spatial distribution characteristics. The proper motion field of the young stars shows a complex structure with spatially coherent patterns. NGC 346's upper main sequence and pre-main sequence stars follow very similar motion patterns, with the outer parts of the cluster being characterized both by outflows and inflows. The proper motion field in the inner ~10 pc shows a combination of rotation and inflow, indicative of inspiraling motion. The rotation velocity in this regions peaks at ~3 km s-1, whereas the inflow velocity peaks at ~1 km s-1. Subclusters and massive young stellar objects in NGC 346 are found at the interface of significant changes in the coherence of the proper motion field. This suggests that turbulence is the main star formation driver in this region. Similar kinematics observed in the metal-poor NGC 346 and in the Milky Way's star-forming regions suggest that the differences in the cooling conditions due to different amounts of metallicity and dust density between the SMC and our galaxy are too small to alter significantly the process of star cluster assembly and growth. The main characteristics of our findings are consistent with various proposed star cluster formation models.