Let $R$ be a local ring (commutative, Noetherian, over an algebraically closed field; if needed Henselian). Suppose one wants to prove some statement.

Suppose $R$ happens to be the ring of "functions", each of which being defined in a small enough neighborhood of the origin, then one can consider some representative of $Spec(R)$, with genuine geometric points off the origin. And to prove the statement pointwise, by going over all the points. Sometimes, this is simpler than to do the proof in the general case of an "abstract" local ring.

e.g. Suppose we are given two ideals $I,J\subset R$ (defined in a complicated way). And we want to check that the ideal they generate together contains a power of the maximal ideal. Geometrically (if $R$ is an analytic ring) this means: the two subschemes $V(I)$, $V(J)\subset Spec(R)$ intersect at the origin only. Then one can just go over the points in the punctured neighborhood of the origin and to check pointwise. This might greatly simplify the proof. (At least the idea of the proof. At least for some people.)

But, when working with complete local rings, one cannot speak of the "points near the origin", etc.

${\bf Question:}$ Is there some analogue of Lefschetz principle, when working with local rings? Something like: if a statement is formulated over an arbitrary local ring, and can be proven for analytic rings (i.e. $k \{ x_1,..,x_n \} /I$), then it is true for an arbitrary local ring (at least henselian, over $k=\bar{k}$)?

${\bf upd:}$ In view of the comments, an additional example might be helpful. Consider a matrix $A$ over a local ring. Suppose one wants to re-derive the standard upper bound on the height of the fitting ideal $I_j(A)$. The bound is well known in alg.com. But if this matrix happens to be a matrix of "functions", that are computable in some open neighborhood of the origin, then one can derive this bound also geometrically, as the codimension of the corresponding degeneracy locus. The question is: after I prove geometrically some bound of this type, which invocations should be pronounced to ensure the validity over an arbitrary local ring?

$\bf upd2:$ While the general approach to treat such questions uses the model theory (as is explained below), the following standard tricks often help.

Suppose a statement is known over some ring R. Often it trivially follows for any quotient of R and any subring of R.

Suppose the statement involves a finite amount of matrices, finitely generated ideals, modules. Then instead of the whole ring R take the subring generated by all the entries. This subring often embeds into some "good" ring.

(If nothing else works) Suppose there exists a counterexample to the statement, over some "bad" ring. Starting from this counterexample try to construct a counterexample over the "good" ring. (e.g. cut all the tails in the Taylor expansions, to get polynomials)