Let $k$ be an algebraically closed field of characteristic $p>0$. Let $W(k)$ be the ring of Witt vectors with coefficients in $k$. A motivic conjecture of Milne relates, in the case of abelian schemes over $W(k)$, the \'etale cohomology with $\Bbb Z_p$ coefficients to the crystalline cohomology with coefficients in $W(k)$. In this talk we report on the proof of this conjecture in the more general context of $p$-divisible groups over $W(k)$ endowed with arbitrary families of crystalline tensors. If $V$ is a discrete valuation ring which is a finite extension of $W(k)$ of index of ramification $e>1$, we provide examples which show that the conjecture is not true in general over $V$ and we also mention some general cases in which the conjecture does hold over $V$. Our results extend previous results of Faltings. As a main new tool we construct global deformations of $p$-divisible groups endowed with crystalline tensors over certain regular, formally smooth schemes over $W(k)$ whose special fibers over $k$ have a Zariski dense set of $k$-valued points.