  The Fornax cluster provides an unparalleled opportunity to investigate in
detail the formation and evolution of early-type galaxies in a dense
environment. We aim at kinematically characterizing photometrically detected
globular cluster (GC) candidates in the core of the cluster. We used the
VLT/VIMOS spectroscopic data from the FVSS survey in the Fornax cluster,
covering one square degree around the central massive galaxy NGC 1399. We
confirmed a total of 777 GCs, almost doubling the previously detected GCs,
using the same dataset by Pota et al. (2018). Combined with previous literature
radial velocity measurements of GCs in Fornax, we compiled the most extensive
spectroscopic GC sample of 2341 objects in this environment. We found that red
GCs are mostly concentrated around major galaxies, while blue GCs are
kinematically irregular and are widely spread throughout the core region of the
cluster. The velocity dispersion profiles of blue and red GCs show a quite
distinct behaviour. Blue GCs exhibit a sharp increase in the velocity
dispersion profile from 250 to 400km/s within 5 arcminutes (29 kpc/1 reff of
NGC 1399) from the central galaxy. The velocity dispersion profile of red GCs
follows a constant value in between 200-300km/s until 8 arcminutes (46
kpc/1.6reff), and then rises to 350km/s at 10 arcminutes (58 kpc/2 reff).
Beyond 10 arcminutes and out to 40 arcminutes (230 kpc/8 reff), blue and red
GCs show a constant velocity dispersion of 300+/-50km/s, indicating that both
GC populations are tracing the cluster potential. We have kinematically
confirmed and characterized the previously photometrically discovered
overdensities of intra-cluster GCs. We found that those substructured
intra-cluster regions in Fornax are dominated mostly by blue GCs.
