Given fibration sequences $F\rightarrow E\rightarrow B$ and $F'\rightarrow E'\rightarrow B'$, consider the homology Serre spectral sequence $S$ for the product of fibrations $F\times F'\rightarrow E\times E'\rightarrow B\times B'$.
Under the ideal conditions on our spaces (torsion-free homology to be safe) I would guess that $S$ has the expected behaviour, namely with differentials being derivations given in terms of the differentials in the spectral sequences for each of the individual fibrations in the product.
I haven't been able to find a single reference, even in McCleary's book. Unless I didn't look hard enough, or this is obvious from the construction of the spectral sequence. Anybody have any idea?