Main Question:What Is the correpondence between flows and vector fields in algebraic geometry?

Here is a more precise statement could be an answer If it was true (I have no idea it is):

"Proposition": Let $X$ be anice enough(intentionally ambiguous) scheme over a field $k$ with tangent sheaf $\mathcal{T}_X$. For every point $x \in X$ There is a 1-1 correpsodence between germs of vector fields through $x$ and "germs" of formal flows.More precisely for every $v \in \mathcal{T}_{X,x}$ there corresponds a (continuous) morphism of complete local rings $\phi_v \in Hom_{cont}(\widehat{\mathcal{O}_{X,x}},\widehat{\mathcal{O}_{X,x} \otimes_k k[[t]]})$ and vice versa.

This "Proposition" was mentioned to me as a brief comment by a very respected mathematician (who's name I won't mention since I'm not completely convinced that the above is a fiathfull interpretation of his original statement). Unfortunately he didn't have time to elaborate on this and couldn't point me to a relevant reference.

Reference request: What are some good sources to turn to considering questions about formal geometry? Specifically the formulation of the classical theory of differential equations (of which this question is a particular case) in formal geometric terms.

Even the "proposition" above isn't really precise since I'm not sure what meaning I should give to the tensor product (some kind of completed tensor product perhaps). Which leads me to the following minor question.

Consider the category whose objects are topological rings $A$ whose topology can be generated by an $I$-adic filtration for some ideal $I \subset A$ and let the morphisms be continuous homomorphisms of rings.

(minor) Question:For arbitrary $A$ and $B$ in this category what is pushout of $A \leftarrow \mathbb{Z} \to B$? should this be the correct interpretation of the tensor product above?