This paper is devoted to the classification of the upper motives of all the products of projective linear groups with coefficients in a finite field. We~first give a complete classification of the upper motives of all the projective linear groups and derive from it the motivic dichotomy. We then provide several classification results as well as counterexamples for arbitrary products of projective linear groups, showing that the situation is less rigid. The proofs involve a neat study of rational maps between generalized Severi-Brauer varieties which is certainly of independent interest.