  Given $k\ge 2$ and two $k$-graphs ($k$-uniform hypergraphs) $F$ and $H$, an
$F$-factor in $H$ is a set of vertex disjoint copies of $F$ that together cover
the vertex set of $H$. Lenz and Mubayi were first to study the $F$-factor
problems in quasi-random $k$-graphs with a minimum degree condition. Recently,
Ding, Han, Sun, Wang and Zhou gave the density threshold for having all
$3$-partite $3$-graphs factors in quasi-random $3$-graphs with vanishing
minimum codegree condition $\Omega(n)$.
  In this paper, we consider embedding factors when the host $k$-graph is
$k$-partite and quasi-random with partite minimum codegree condition. We prove
that if $p>1/2$ and $F$ is a $k$-partite $k$-graph with each part having $m$
vertices, then for $n$ large enough and $m\mid n$, any $p$-dense $k$-partite
$k$-graph with each part having $n$ vertices and partite minimum codegree
condition $\Omega(n)$ contains an $F$-factor. We also present a construction
showing that $1/2$ is best possible. Furthermore, for $1\leq \ell \leq k-2$, by
constructing a sequence of $p$-dense $k$-partite $k$-graphs with partite
minimum $\ell$-degree $\Omega(n^{k-\ell})$ having no $K_k(m)$-factor, we show
that the partite minimum codegree constraint can not be replaced by other
partite minimum degree conditions. On the other hand, we prove that $n/2$ is
the asymptotic partite minimum codegree threshold for having all fixed
$k$-partite $k$-graph factors in sufficiently large host $k$-partite $k$-graphs
even without quasi-randomness.
