Let $X$ be a $K$-scheme (affine if necessary), $K|k$ an extension of fields (separable, finite, … as necessary).
Then $k \hookrightarrow K$ is unique as a homomorphism of $k$-algebras and so $\operatorname{Spec} K \to \operatorname{Spec} k$ is unique as a morphism of $k$-schemes.
In this way, $X$ has the unique associated structure morphism $X \to \operatorname{Spec} K \to \operatorname{Spec} k$, which makes $X$ into a $k$-scheme.
Is this construction helpful for anything or do you know of a place where it is used?
I have not seen it used so far, but am new to the language of schemes and may be missing something.
Best Answer
This isn't widely used and I also haven't seen it around. But it can lead to some weird things which play with intuition, atleast for me.
Suppose $ X $ is the hyperbola $ \operatorname{Spec} \mathbb{C} [u,v] / (uv -1) $ considered as a scheme over $ \operatorname{Spec} \mathbb{C} $. We can consider it as a scheme $ X|_{ \mathbb{R} } $ over $ \operatorname{Spec} \mathbb{R} $ via the inclusion of $ \mathbb{R} $ in $ \mathbb{C} $. Then $ X|_{ \mathbb{R} } $ has no real points (going against the intuition that the hyperbola has lots of real points) simply because there is no morphism $ \mathbb{C} [u,v] / (uv -1) \rightarrow \mathbb{R} $ over $ \mathbb{R} $.
Also, base changing $ X|_{ \mathbb{R} } $ back to $ \mathbb{C} $ does not give you $ X $, but two disjoint copies of $ X $.
A better concept of 'restriction of scalars' for schemes over fields is Weil restriction. You might want to look that up.