What is the appropriate functoriality statement of a filtered chain map between filtered spectral sequences?

As in, say we have a filtered chain map f, between spectral sequences of filtered chain complexes, f:C→C′, then what map does this induce on each page?

What relation, if any, might there be between these induced maps, and the map that f induces on the total homology of the complexes f∗:H(C)→H(C′)?

  Is there a way that the associated graded of the map f is related to the above?

Can you please suggest a place to read about these issues?

Thank you!

What is the appropriate functoriality statement of a filtered chain map between filtered spectral sequences?

Suppose that we have two filtered chain complexes $C,C'$ and a filtered chain map $f\colon C\to C′$. This induces a map between the spectral sequences of the filtered chain complexes, and my question is what map does this induce on each page.

In particular, what relation, if any, is there between these induced maps, and the map that $f$ induces on the total homology of the complexes $f_\ast\colon H(C)\to H(C′)$? Is there a way that the associated graded of $f$ is related to the above? Is there any prefered reference for these matters?