This is part of exercise 1.11 in Mazza, Voevodsky, Weibel's Lecture Notes on Motivic Cohomology.
Here's some notational stuff: we're working in the category $Cor_k$ whose objects are the smooth separated $k$ schemes and whose morphisms are finite correspondences.
Let $x$ be a closed point on $X$ considered as a correspondence from $S = Spec \, k$ to $X$. Show that the composition $S \rightarrow X \rightarrow S$ is multiplication by the degree $[k(x):k]$.
To the best of my knowledge, the question should be interpreted as follows: The group $Cor(X,S)$ is isomorphic to the free abelian group $\mathbb Z$ generated by the elementary correspondence $X$ in $X \times S$. We compose a fixed correspondence with this group, and we want to identify the map $\mathbb Z \rightarrow \mathbb Z$. The closed point $x$ is interpreted as a correspondence as follows: there's an inclusion $k \hookrightarrow k(x)$ giving a map of smooth schemes $x \rightarrow S$. We take the transpose of the graph of this morphism to get a correspondence in $Cor(S,x)$, then post compose with the inclusion $x \hookrightarrow X$ to get a correspondence in $Cor(S,X)$.
My problem seems to be coming from actually calculating this correspondence (I think the next part, composing with the graph of $X \rightarrow S$ is fine). Assume for simplicity that $k(x)/k$ is a Galois extension. Here's my attempt at the computation:
The transpose of the graph mentioned above is the correspondence $S \times x$ in $Cor(S,x)$. The graph of the inclusion is $x \times x$ in $Cor(x,X)$. Let $[T] = (S \times x \times X) \cdot (S \times x \times x)$. This intersection is proper with multiplicity one, so we get $[T] = [S \times x \times x]$ in $S \times x \times X$. When I push forward $[T]$ to $S \times X$, I need to know the degree of the map $x \times_S x \rightarrow x$. Now $x \times_S x = Spec \, k(x) \otimes_k k(x)$. Let $L = k(x)$, the residue field at the closed point $x$. By assumption that $L$ is Galois over $k$, we get the chain of isomorphisms $L \otimes L \cong k[x]/p(x) \otimes L \cong L[x]/(p(x)) \cong L[x]/(x-a_1) \oplus \cdots \oplus L[x]/(x-a_n) \cong L^n$ where $n$ is the degree of the extension $[L:k]$. Hence when I take Spec, I get $n$ disjoint copies of $Spec \, L$. So this correspondence is already $[k(x):k]\{x\}$ in $Cor(S,X)$.
Now, when I go to calculate the composition of this correspondence with the graph of $X \rightarrow S$, I pick up another factor of $[k(x):k]$, which gives me the wrong answer.
Here's an easier way.
We want to show the composition of correspondences $S\rightarrow X \rightarrow S$ is the degree $[k(x):k]$ in $Cor_k(S,S)$.
Considering the closed point $x$ as the correspondence $[S\times x]\in Cor_k(S,X)$ and the morphism $X\rightarrow S$ given by the generator $[X\times S]\in Cor_k(S,X)$. Take their intersection product $[S\times x]\cdot [X\times S]$, which is $[S\times x \times S]$. Now we just need to show the pushforward map $p:S\times x \times S\rightarrow S\times S$ is of degree $[k(x):k]$. But we can just use the isomorphisms $S\times x \times S\cong S\times x$ and $S\times S\cong S$ to see $p$ is isomorphic with the first projection $S\times x\rightarrow S$. That the degree agrees here is clear.