  We study the nonlinear embeddability of Banach spaces and the
equi-embeddability of the family of Kalton's interlaced graphs $([\mathbb
N]^k,d_{\mathbb K})_k$ into dual spaces. Notably, we define and study a
modification of Kalton's property $\mathcal Q$ that we call property
$\mathcal{Q}_p$ (with $p \in (1,+\infty]$). We show that if $([\mathbb
N]^k,d_{\mathbb K})_k$ equi-coarse Lipschitzly embeds into $X^*$, then the
Szlenk index of $X$ is greater than $\omega$, and that this is optimal, i.e.,
there exists a separable dual space $Y^*$ that contains $([\mathbb
N]^k,d_{\mathbb K})_k$ equi-Lipschitzly and so that $Y$ has Szlenk index
$\omega^2$. We prove that $c_0$ does not coarse Lipschitzly embed into a
separable dual space by a map with distortion strictly smaller than
$\frac{3}{2}$. We also show that neither $c_0$ nor $L_1$ coarsely embeds into a
separable dual by a weak-to-weak$^*$ sequentially continuous map.

Functional Analysis;Metric Geometry