  Let $k$ be a number field. In the spirit of a result by Yongqi Liang, we
relate the arithmetic of rational points over finite extensions of $k$ to that
of zero-cycles over $k$ for Kummer varieties over $k$. For example, for any
Kummer variety $X$ over $k$, we show that if the Brauer-Manin obstruction is
the only obstruction to the Hasse principle for rational points on $X$ over all
finite extensions of $k$, then the ($2$-primary) Brauer-Manin obstruction is
the only obstruction to the Hasse principle for zero-cycles of any given odd
degree on $X$ over $k$. We also obtain similar results for products of Kummer
varieties, K3 surfaces and rationally connected varieties.
