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!