  Let $\nu$ be a place of a global function field $K$ over a finite field, with
associated affine function ring $R_\nu$ and completion $K_\nu$, and let $1 \leq
\mathfrak{m}<\textbf{d}$. The aim of this paper is to prove an effective triple
joint equidistribution result for primitive partial $R_\nu$-lattices $\Lambda$
of rank $\mathfrak{m}$ in $K_\nu^{\;\textbf{d}}$ as their covolume tends to
infinity: of their $K_\nu$-linear span $V_\Lambda$ in the rank-$\mathfrak{m}$
Grassmannian space of $K_\nu^{\;\textbf{d}}$; of their shape in the modular
quotient by $\operatorname{PGL}_\mathfrak{m}(R_\nu)$ of the Bruhat-Tits
buildings of $\operatorname{PGL}_\mathfrak{m}(K_\nu)$; and of the shape of
$\Lambda^\perp$ in the similar quotient for
$\operatorname{PGL}_{\textbf{d}-\mathfrak{m}}(K_\nu)$, where $\Lambda^\perp$ is
the orthogonal partial $R_\nu$-lattice of rank $\textbf{d}-\mathfrak{m}$ in the
dual space of $K_\nu^{\;\textbf{d}}$. The main tools are a new refined
$\text{LU}$ decomposition by blocks of elements of
$\operatorname{SL}_\textbf{d}(K_\nu)$, techniques of Gorodnik and Nevo for
counting integral points in well-rounded families of subsets of algebraic
groups, and computations of volumes of various homogeneous spaces associated
with partial $R_\nu$-lattices.
