  Given an extension of number fields $E \subset F$ and a projective variety
$X$ over $F$, we compare the problem of counting the number of rational points
of bounded height on $X$ with that of its Weil restriction over $E$. In
particular, we consider the compatibility with respect to the Weil restriction
of conjectural asymptotic formulae due to Manin and others. Using our methods
we prove several new cases of these conjectures. We also construct new
counterexamples over every number field.
