Let $K$ be a field, $\partial_i$ be commuting derivations on $K$, and consider the ring $R=K[\partial_1\ldots \partial_n]$ (it is implicitly assumed that the derivations do not commute with elements of $K$ and the commutation relation is the obvious $\partial_i k - k\partial_i=\partial_i(k)$).

For example, $R$ can be the ring of differential operators with rational function coefficients, etc.

It appears that much of the theory of Groebner bases for polynomial rings $S=K[x_1\ldots x_n]$ can be trivially restated for $R=K[\partial_1\ldots \partial_n]$, since when we work with a monomial order, we usually track only what happens to the leading monomial, and it is the same for $R$ and $S$. I am pretty sure that this is also true for modules over $R$. (If I am wrong and this is not true, an explanation of this will be a perfect answer to this question:) )

Is there a reference which restates the standard results over $S$ for modules over $R$?

Most of the references I have found deal with the more(?) complicated case of polynomial coefficients. Other than that, I am aware of two references, both by Nobuki Takayama, "Gröbner basis and the problem of contiguous relations" and a chapter in "Gröbner Bases: Statistics and Software Systems". The first basically introduces a version of Buchberger's algorithm for modules over $R$ and the second briefly goes over the very basic facts for ideals in $R$.