The Leray spectral sequence of a continuous map $f : X \to Y$ between two topological spaces can be constructed, as far as I understand, in two ways, one very general, and the other a bit more restrictive.

For the first definition, we look at the direct image functor $f_* : Sh(X) \to Sh(Y)$, between the categories of sheaves of abelian groups, and use Grothendieck spectral sequences.

For the second definition, we assume that $f$ is a smooth map between smooth manifolds, and we define a spectral sequence by means of a double complex, based on the Cech complex associated with an open covering of $Y$, and the complex of differential forms on $X$.

On the other hand, Serre was able to define, without using the concept of sheaves, the Serre (or Leray-Serre) spectral sequence in the case where $f$ is a fibration and $Y$ has some nice properties. Introducing a filtration on the singular cohomology groups $H^*(X)$ by means of the skeleton of $Y$, he could build successively the $E_r$ terms of a spectral sequence, converging to $H^*(X)$. This provides a very "visual" understanding of this spectral sequence.

A nice feature of the Serre spectral sequence is that, for any other fibration $f' : X' \to Y'$, the spectral sequences of $f$ and $f'$ will be related as soon as there is a commutative diagram $\require{AMScd}$ $$\begin{CD} H^*(Y) @>>> H^*(Y')\\ @V f^* V V @VV f'* V\\ H^*(X) @>>> H^*(X') \end{CD}.$$ for which the horizontal maps preserve the filtration. In particular, the horizontal maps do NOT need to come from maps of spaces.

Conversely, if we wanted to relate the spectral sequences of $f$ and $f'$ using Grothendieck's setting, I assume that we would need to have a commutative diagram $$\begin{CD} Sh(X) @>>> Sh(X')\\ @V f_* V V @VV f'_* V\\ Sh(Y) @>>> Sh(Y') \end{CD}.$$ In this case, I don't see how we could obtain horizontal maps which would not come from maps of spaces.

I guess that my question could be: is it possible to define a map between the Leray spectral sequences of continuous maps $f$ and $f'$, without requiring that it comes from a map of spaces ? For instance, could it come from a map in cohomology, preserving some kind of filtration similar to that of Serre spectral sequence ?