  In this note we study the functoriality of the coniveau filtration in motivic
homotopy theory via a moving lemma over a base scheme, extending previous works
of Levine and Bachmann-Yakerson. The main result is that the motivic stable
homotopy category can be modeled on a smaller site, the "smooth-smooth site".
The proof is based on a new approach to the purity theorem of Morel-Voevodsky
using specialization maps, which turns out to hold even in absence of the
$\mathbb{A}^1$-homotopy invariance property. Applications to the homotopy
coniveau tower and to higher Chow-Witt groups are given.
