At the time when I constructed those maps, the motivic spectral sequence was not known to exist. I gave a conjectural construction for that spectral sequence in an American Journal of Math paper called "Relative Cycles and Algebraic K-Theory", but I still don't know whether the gaps in that construction can be (or have been) filled.
I stopped working on this long enough ago that I could be remembering wrong, but my best recollection is that if the conjectural construction in RCAKT can be made to work, and if the spectral sequence it yields is the same as Friedlander-Suslin's (which it certainly ought to be), then you can piece together results in those two papers to show that the maps in question are compatible with the differentials. (You might also need some results from my paper "Some Filtrations on Higher K-Theory and Related Invariants".)
There might also be a much easier argument, directly using the Friedlander-Suslin construction, which would save you from mucking around in those other papers.
I apologize for many things, including my hazy memory, my failure to state this result explicitly in the original papers, and the fact that I'm not up to date on the current state of the art.