Recall that a morphism of rings $R\to S$ is called (essentially) _smooth_ if it is formally smooth and (essentially) finitely presented.

(Note: $R\to S$ is _essentially finitely presented_ provided that $S$ is the localization of some finitely  
 presented $R$-algebra $T$ at some multiplicative system $A \subset T$, that is, $S=A^{-1}T$.)

In class, our professor said that working with smooth or essentially smooth morphisms yields an effectively equivalent theory.  This motivates my question: Is there a general way to lift results from the smooth case to the essentially smooth case?

For instance, if $R$ is reduced, and $R\to S$ is smooth, then $S$ is reduced.  It turns out that the same fact is true for essentially smooth maps.  Is there a good way to "lift" this result to the more general case?