Yongqi Liang has shown that for rationally connected varieties over a number field K, sufficiency of the Brauer-Manin obstruction to the existence of rational points over all finite extensions of K implies sufficiency of the Brauer-Manin obstruction to the existence of zero-cycles of degree 1 over K. I will discuss joint work with Francesca Balestrieri where we extend Liang's result to Kummer varieties.