In this paper we prove the birational rigidity of Fano-Mori fibre spaces $\pi\colon V\to S$, every fibre of which is a Fano complete intersection of index 1 and codimension $k\geqslant 3$ in the projective space ${\mathbb P}^{M+k}$ for $M$ sufficiently high, satisfying certain natural conditions of general position, in the assumption that the fibre space $V/S$ is sufficiently twisted over the base. The dimension of the base $S$ is bounded from above by a constant, depending only on the dimension $M$ of the fibre (as the dimension of the fibre $M$ grows, this constant grows as $\frac12 M^2$).