  The relation between the free Banach lattice generated by a Banach space and
free dual spaces is clarified. In particular, it is shown that for every Banach
space $E$ the free $p$-convex Banach lattice generated by $E^{**}$, denoted
$FBL^p[E^{**}]$, admits a canonical isometric lattice embedding into
$FBL^p[E]^{**}$ and $FBL^p[E^{**}]$ is lattice finitely representable in
$FBL^p[E]$. Moreover, we also show that for $p>1$, $FBL^p[E]^{**}$ can actually
be considered as the free dual $p$-convex Banach lattice generated by $E$,
whereas for $p=1$ this happens precisely when $E$ does not contain complemented
copies of $\ell_1$.
