Topological K-theory and characteristic classes of module bundles

Question

Let $R$ be a commutative ring and let $A$ be a commutative unital topological $R$-algebra. By means of replacing vector spaces with $A$-modules, one can define $A$-module bundle in analogy to vector bundles.

1. Is there some version of topological K-theory for $A$-module bundles?
2. Is there some version of Serre-Swan theorem in this context, relating some projective $A$-modules with $A$-module bundles? In addition, if the topological K-theory for a $A$-module bundle is well-defined, is it related with algebraic K-theory of $A$?
3. Is there some version of characteristic classes for $A$-module bundles? If the topological K-theory makes sense, can these characteristic classes be regarded as natural transformations from K-theory to some other cohomology theory?

Thank you.

P.S: about K-theory for $A$-module bundles, I found this paper https://link.springer.com/article/10.1007/s12215-008-0004-9

P.S: In my context I found the structure of a $R[G]$-module bundle on an object of my interest, where $R[G]$ is a group algebra. I would like to study invariants of it.