For a ${\mathbb Z}$-filtered ring ${\mathbb k}$ one can consider the category ${\mathbb k}\text{-filt}$ of ${\mathbb Z}$-filtered ${\mathbb k}$-modules, equipped with the exact structure which declares a sequence $X\to Y\to Z$ to be exact if $0\to X_k\to Y_k\to Z_k\to 0$ is exact for all $k$.
As for any exact category, this gives rise to a derived category $\textbf{D}({\mathbb k}\text{-filt})$ of complexes of ${\mathbb Z}$-filtered ${\mathbb k}$-modules.
Questions:
Is $\textbf{D}(A\text{-filt})$ a compactly or at least well-generated triangulated category?
What is known about model categorical enhancements for $\textbf{D}({\mathbb k}\text{-filt})$?
In particular I would like to understand if, given a morphism of ${\mathbb Z}$-filtered rings ${\mathbb k}^{\prime}\to{\mathbb k}$, there is a left adjoint to the forgetful functor $\textbf{D}({\mathbb k}\text{-filt})\to\textbf{D}({\mathbb k}^{\prime}\text{-filt})$, and, if yes, how it looks like explicitly.
Generally I'm interested in any source treating derived categories of filtered modules; so far I have only found literature on derived categories of exact categories in general, not focussing on this special case, however.
A model-categorical enhancement of D(k-filt) is constructed in arXiv:1602.01515, see Corollary 3.58 there. It also shows that this model category is combinatorial and compactly generated, as requested.
A morphism f: k→k' of Z-filtered rings induces a Quillen adjunction, as shown in Theorem 5.5 there. In fact, if f is a graded equivalence, then the Quillen adjunction is a Quillen equivalence, as mentioned there.
Explicitly, the (derived) left adjoint to the forgetful functor from filtered k'-modules to filtered k-modules can be explicitly described as the (derived) filtered tensor product (i.e., the tensor product in the monoidal category of filtered objects) with k' (as a k-module).