Behaviour of the Serre spectral sequence on a product of fibrations Remark 15.4 in Switzer's Algebraic topology: Homology and homotopy talks about constructing the internal product on a Serre spectral sequence by passing first through the external product you describe. (He also doesn't finish the job, which he decries as extremely tedious.) There might also be something in the references of this recent MO question: mathoverflow.net/questions/225579/… .