If $f:X_0\rightarrow X$ is a closed immersion of locally noetherian schemes such that the topological spaces of $X_0$ and $X$ are identical (or, more generally, if $f$ is a universal homeomorphism), then it is known (see e.g. SGA1, Thm I.8.2, or, more generally, SGA1, Thm. IX.4.10) that pullback along $f$ induces an equivalence between the categories of etale $X$-schemes and etale $X_0$-schemes.

The main property of étale morphisms is the "infinitesimal lifting criterion", after which the definition of formally étale morphisms is modeled.

Is a "topological invariance result" as above also true for the categories of formally étale $X_0$- and $X$-schemes?