  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.
