This is a probably stupid question, and I guess the answer is no, but it's worth trying. Consider the spectral sequence $E^{pq}_r$ associated to a bicomplex (of abelian groups). At page $r$ we will have several parallel complex with differentials $\delta_r$. Is it possible to define $\tilde{\delta}_{r+1}$ at page $r$ as so that it defines maps of $\delta_r$-chain complexes? We also ask that $\tilde{\delta}_{r+1}$ induces $\delta_{r+1}$ in homology.
If this is true, what if we start with a multicomplex?