I am trying to understand the conditions for two Leray spectral sequences to be related by a map.

Let $f_1 : X_1 \to Y_1$ and $f_2 : X_2 \to Y_2$ be two continuous maps of topological spaces (with CW-complex structure if this can help).

I am wondering under what condition we could relate the Leray spectral sequences of $f_1$ and $f_2$.

For instance, it is explained in McLeary's users guide that, if $f_i$ is a fibration, the Leray spectral sequence of $f_i$ can be constructed from the filtration of cochain complexes $$F^pC^*(X_i) := \ker C^*(X_i, \mathcal{R}) \to C^*(f_i^{-1}(Y_i^p), \mathcal{R}),$$ where $Y_i^p$ denotes the $p$-skeleton of $Y_i$, and $\mathcal{R}$ is a commutative ring. This filtration induces a spectral sequence, called de Leray-Serre spectral sequence, which converges to $H^*(X_i, \mathcal{R})$.

In such a case, any map $g : C^*(X_2, \mathcal{R}) \to C^*(X_1, \mathcal{R})$ preserving the filtration would induce a map between the Leray-Serre spectral sequences of $f_1$ and $f_2$.

My question is: does there exist a similar condition in the general case of continuous maps $f_1$ and $f_2$ ? For instance, would it be possible to define the Leray spectral sequence based on the same filtration $F^p$ ? In this case, the same condition as above would still hold.

Thanks in advance