In this paper, we consider "diagrams of rings", or functors from a small category to the category of rings, and the corresponding diagrams of groups $K_i.$ Classically, this was initiated by Milnor. The main result of this paper is the direct comparison of the filtration in classical algebraic $K$-theory discussed in J. Duflot, "Simplicial groups that are models for algebraic $K$-theory," Manuscripta Math. 113 (2004), no. 4, 423–470 and J. Duflot and C.T. Marak, "A filtration in algebraic $K$-theory," J. Pure Applied Algebra 151 (2000), no. 2, 135–162 to a corresponding filtration in the Bousfield-Kan spectral sequence associated to a Tot-tower of simplicial groups attached to the diagram of rings.
Jeanne Duflot. "The algebraic $K$-theory of a diagram of rings." Homology Homotopy Appl. 10 (2) 13 - 58, 2008.