Let $p\colon E\to B$ and $p'\colon E'\to B'$ be two fibrations. Assume for simplicity that $B,B'$ are simply connected. Let we have morphism of these fibrations, i.e. two continuous maps $$f\colon E\to E', g\colon B\to B'$$ such that $p'\circ f=g\circ p$. Let $F$ be the fiber of $p$ over some fixed point $b\in B$, and $F'$ be the fiber of $p'$ over $g(b)$.

Question. Does there exist a "natural" morphism of the Leray spectral sequences for the homology, which, in particular, satisfies

1. on the second terms this moprhism $H_*(F)\otimes H_*(B)\to H_*(F')\otimes H_*(B')$ coincides with $f_*\otimes g_*$.

2. $f_*\colon H_*(E)\to H_*(E')$ preserves the Leray filtrations and the associated graded map coincides with the required map on the $\infty$-terms.

A reference would be helpful.