  By a classical result of Roitman, a complete intersection $X$ of sufficiently
small degree admits a rational decomposition of the diagonal. This means that
some multiple of the diagonal by a positive integer $N$, when viewed as a cycle
in the Chow group, has support in $X\times D\cup F\times X$, for some divisor
$D$ and a finite set of closed points $F$. The minimal such $N$ is called the
torsion order. We study lower bounds for the torsion order following the
specialization method of Voisin, Colliot-Th\'el\`ene and Pirutka. We give a
lower bound for the generic complete intersection with and without point.
Moreover, we use methods of Koll\'ar and Totaro to show lower bounds for the
very general complete intersection.
