With your definitions, assuming you mean the configuration space of distinct points, and that the inclusions need to be compatible with the actual locations of the points in some way, there is not such a map for homotopy reasons.
I will write $C(X,k)$ for the unordered configuration space.
Let $X$ be the circle $S^1$. Then $F(X,k)$ is homotopy equivalent to a union of $(k-1)!$ copies of the circle, and $C(X,k)$ is a single circle. The quotient map to $F(X,k)$ is a covering map of degree $k!$; each component of $F(X,k)$ is maps to $C(X,k)$ with degree $k$.
There is an inclusion $C(X,1)$ to $C(X,k)$ for each $k$; it's easy to construct it explicitly, and it is a map of degree $k$ as a map from one circle to another.
There is not, however, a map from $C(X,2)$ to $C(X,3)$ that is compatible with the point positions. Such a map would have to be a covering map of "degree $3/2$", which is impossible.