The Lefschetz principle can be understood in scheme theoretic terms in the following way:
suppose that $X \to S$ is a scheme over a base $S$ (possibly with extra data) which is fppf over $S$. Then we may descend $X$ to $X_0 \to S_0$ where $S_0$ is finite type over $\mathbb Z$. (Here "descend" means that there is a map $S\to S_0$ so that $X$ is recovered from $X_0$ via base-change. For a proof/explanation, search for discussions of "removing Noetherian hypotheses" online. The standard reference is somewhere in EGA IV.)
Now suppose that $P$ is a property that can be checked after faithfully flat base-change; then we use the above method to tranfer $P$ from the context of complex scalars to any field of char. zero.
E.g. if $(X,\mathcal L)$ is a smooth projective variety over a field $k$ of char. zero, then via the above we may descend $(X,\mathcal L)$ to $(X_0,\mathcal L_0)$ over a finite type $\mathbb Z$-scheme $S_0$. The morphism Spec $k \to S_0$ factors as Spec $k \to $ Spec $k_0 \to S_0$, where $k_0$ is a finitely generated subfield of $k$, since $S_0$ is finite type over $\mathbb Z$. Base-changing to $k_0$, we get $(X_0',\mathcal L_0')$ over $k_0$ which recovers $(X,\mathcal L)$ after base-changing to $k$.
Now choose an embedding $k \to \mathbb C$, as we may do since $k_0$ is finitely generated. Base-change to $\mathbb C$ gives $(X',\mathcal L')$.
So we have $(X,\mathcal L)$ and $(X',\mathcal L')$ over $k$ and over $\mathbb C$, both of which are base-changed from $(X_0',\mathcal L_0')$ over $k_0$.
Using the fact that properness, smoothness, and ampleness may be checked after a faithfully flat base-change (in our case, just a change of base field), and are also preserved by such a base-change, and also that formation of the canonical bundled, and of cohomology, also commutes with change of base field, we can transfer Kodaira embedding from $(X',\mathcal L')$ to $(X_0',\mathcal L_0')$, and finally to $(X,\mathcal L)$, as desired.
Note: The fact that $X\to S$ can be recovered from $X_0\to S_0$ is one way of encoding Lefschetz's intiuition that an algebraic variety only requires a finite amount of data to encode, which is what underlies the Lefschetz principle. In practice, people use this a lot, whereas I've never seen anyone use a logical or model-theoretic formulation of the Lefschetz principle in an algebraic geometry argument.
People also use the closed points of $S_0$, which have positive characteristic, to deduce facts about the original $X$ --- thus the decomposition theorem for perverse sheaves in char. zero was first proved by such reduction to char. $p$ methods, as was the bend-and-break lemma in the theory of birational geometry. Raynaud gave a proof of Kodaira embedding by proving it first in a char $p$ setting and then passing to char. $0$ by these methods. In the context of passing from char. $p$ to char. $0$ there also more model-theoretic arguments, such as in some proofs of the Ax-Grothendieck theorem, but my experience in this context too is that "spreading out" arguments (people call the passage from
Spec $k$ of char. zero to $S_0$ "spreading out" over $\mathbb Z$) are much more common.