We investigate the kinematics of a combined sample of 74 globular clusters around NGC 1399. Their high velocity dispersion, increasing with radius, supports their association with the gravitational potential of the galaxy cluster rather than with that of NGC 1399 itself. We find no evidence for rotation in the full sample, although some indication for rotation in the outer regions. The data do not allow us to detect differences between the kinematics of the blue and red sub-populations of globular clusters. A comparison between the globular cluster systems of NGC 1399 and those of NGC 1404 and NGC 1380 indicates that the globular clusters in all three galaxies are likely to have formed via similar mechanisms and at similar epochs. The only property which distinguishes the NGC 1399 globular cluster system from these others is that it is ten times more abundant. We summarize the evidence for associating these excess globulars with the galaxy cluster rather than with NGC 1399 itself, and suggest that the over-abundance can be explained by tidal stripping, at an early epoch, of neighboring galaxies and subsequent accumulation of globulars in the gravitational potential of the galaxy cluster.