  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 $\frac12 M^2$).
