  We compute the leading order connected four-point function of the primordial
curvature perturbation coming from the four-point function of the fields in
multi-field DBI inflation models. We confirm that the consistency relations in
the squeezed limit and in the counter-collinear limit hold as in single field
models thanks to special properties of the DBI action. We also study the
momentum dependence of the trispectra coming from the adiabatic, mixed and
purely entropic contributions separately and we find that they have different
momentum dependence. This means that if the amount of the transfer from the
entropy perturbations to the curvature perturbation is significantly large, the
trispectrum can distinguish multi-field DBI inflation models from single field
DBI inflation models. A large amount of transfer $T_{\mathcal{RS}} \gg 1 $
suppresses the tensor to scalar ratio $r \propto T_{\mathcal{RS}}^{-2}$ and the
amplitude of the bispectrum $f_{NL}^{equi} \propto T_{\mathcal{RS}}^{-2}$ and
so it can ease the severe observational constraints on the DBI inflation model
based on string theory. On the other hand, it enhances the amplitude of the
trispectrum $\tau_{NL}^{equi} \propto T_{\mathcal{RS}}^2 f_{NL}^{equi 2}$ for a
given amplitude of the bispectrum.
