Boundedness of the p-primary torsion of the Brauer group of products of varieties

Let k be a field finitely generated over its prime subfield. We prove that the quotient of the Brauer group of a product of varieties over k by the Brauer groups of factors has finite exponent. The bulk of the proof concerns p-primary torsion in characteristic p. Our approach gives a more direct proof of the boundedness of the p-primary torsion of the Brauer group of an abelian variety, as recently proved by D'Addezio. We show that the transcendental Brauer group of a Kummer surface over k has finite exponent, but can be infinite when k is an infinite field of positive characteristic. This answers a question of Zarhin and the author.