Let $X$ be a variety over a field $k$. I know the definition of $k$-rational point: a closed point $x$ is $k$-rational if its residue field $k(x)$ is equal to $k$ (in general, $k(x)$ is only a finite extension of $k$).
Now assume that $Z$ is a closed subscheme of $X$. Is there a notion of being $k$-rational? If so, how is it define?
I'm wondering how important it is to consider $k$-rational subschemes in certain constructions if we want the output to be again defined over $k$.
For instance, I've seen that people compute the motive of a blow-up $Bl_Z X$ in terms of the motives of $X$ and $Z$. They always work in the category of motives over $k$, but it is never said something like "assume $Z$ is $k$-rational". Does the blow-up $Bl_Z X$ always have a natural structure of variety over $k$?