Let $X,Y,Z$ be varieties. Given two correspondences $\Gamma_1 \subset X \times Y$ and $\Gamma_2 \subset Y \times Z$ there is a composition, $$ [\Gamma_1] \circ [\Gamma_2] = \pi_{13 *} (\pi_{12}^* [\Gamma_1] \cdot \pi_{23}^* [\Gamma_2]) $$ carried out in the Chow ring.

One natural source of correspondences are the graphs of rational maps $f : X \dashrightarrow Y$. Unfortunately, it is not true that, $$ [\Gamma_{f \circ g}] \neq [\Gamma_f] \circ [\Gamma_g] $$ when $f$ and $g$ are not regular. Indeed, let $f, g$ be the Cremona involution $\iota$ of $\mathbb{P}^2$ then $\Gamma_{\iota^2} = \Delta$ and $[\Gamma_\iota] = [\Delta] + [\ell] \times [\ell]$ where $[\ell]$ is the class of a line. Then, $$ [\Gamma_{\iota}] \circ [\Gamma_{\iota}] = [\Delta] + 3 [\ell] \times [\ell] $$
However, we note that the discrepancy is supported on a divisor.

To capture this, let $\mathrm{Spec}(K) \to X$ be the generic point and along the map $X_K \to X \times X$ we get a pullback $r : \mathrm{CH}_d(X \times X) \to \mathrm{CH}_0(X_K)$. Is it true that if $f, g : X \dashrightarrow X$ are birational automorphisms then,

$$ r([\Gamma_{f \circ g}]) = r([\Gamma_f] \circ [\Gamma_g]) $$

This rases another question. Does the composition operation on $\mathrm{CH}_d(X \times X)$ correspond to some well-defined operation on $\mathrm{CH}_0(X_K)$?

A particularly interesting case is when $X$ is rationally connected. Then $\mathrm{ker}(\mathrm{CH}_0(X_K) \xrightarrow{\mathrm{deg}} \mathbb{Z})$ is torsion. I would like to know how the torsion classes of $[\Gamma_f]$ behave with respect to composition.