  We study the assembly of globular clusters (GCs) in 9 galaxy clusters using
the cosmological simulation Illustris. GCs are tagged to individual galaxies at
their infall time. The tidal removal of GCs from their galaxies and the
distribution of the GCs within the cluster is later followed self-consistently
by the simulation. The method relies on the simple assumption of a single
power-law relation between halo mass (M_vir) and mass in GCs (M_GC) as found in
observations. We find that the GCs specific frequency $S_N$ as a function of
V-band magnitude naturally reproduces the observed "U"-shape, due to the
combination of a power law M_GC-M_vir relation and the non-linear M_*-M_vir
relation from the simulation. Additional scatter in the $S_N$ values are traced
back to galaxies with early infall times due to the evolution in the M_*-M_vir
relation with redshift. GCs that have been tidally removed from their galaxies
form today the intra-cluster component from which about ~60% were brought in by
galaxies that orbit today within the cluster potential. The remaining "orphan"
GCs are contributed by satellite galaxies with a wide range of stellar masses
that are fully tidally disrupted at z=0. This intra-cluster component is a good
dynamical tracer of the dark matter potential. As a consequence of the accreted
nature of most intra-cluster GCs, their orbits are fairly radial with a
predicted orbital anisotropy \beta >= 0.5. However, local tangential motions
may appear as a consequence of localized substructure, providing a possible
interpretation to the \beta<0 values suggested in observations of M87.
