  Consider weak approximation for 0-cycles on a smooth proper variety defined
over a number field, it is conjectured to be controlled by its Brauer group.
Let $X$ be a Ch\^atelet surface or a smooth compactification of a homogeneous
space of a connected linear algebraic group with connected stabilizer. Let $Y$
be a rationally connected variety. We prove that weak approximation for
0-cycles on the product $X\times Y$ is controlled by its Brauer group if it is
the case for $Y$ after every finite extension of the base field. We do not
suppose the existence of 0-cycles of degree $1$ neither on $X$ nor on $Y$.
