  We begin by describing the unit ball of the free $p$-convex Banach lattice
over a Banach space $E$ (denoted by ${\mathrm{FBL}}^{(p)}[E]$) as a closed
solid convex hull of an appropriate set. Based on it, we show that, if a Banach
space $E$ has the $\lambda$-Approximation Property, then
${\mathrm{FBL}}^{(p)}[E]$ has the $\lambda$-Positive Approximation Property.
Further, we show that operators $u \in B(E,F)$ (where $E$ and $F$ are Banach
spaces) which extend to lattice homomorphisms from ${\mathrm{FBL}}^{(q)}[E]$ to
${\mathrm{FBL}}^{(p)}[F]$ are precisely those whose adjoints are
$(q,p)$-mixing. Related results are also obtained for free lattices with an
upper $p$-estimate.
