  This is the first paper of a three-part series in which we develop a theory
of conformal blocks for $C_2$-cofinite vertex operator algebras (VOAs) that are
not necessarily rational. The ultimate goal of this series is to prove a
sewing-factorization theorem (and in particular, a factorization formula) for
conformal blocks over holomorphic families of compact Riemann surfaces,
associated to grading-restricted (generalized) modules of $C_2$-cofinite VOAs.
  In this paper, we prove that if $\mathbb V$ is a $C_2$-cofinite VOA, if
$\mathfrak X$ is a compact Riemann surface with $N$ incoming marked points and
$M$ outgoing ones, each equipped with a local coordinate, and if $\mathbb W$ is
a grading-restricted $\mathbb V^{\otimes N}$-modules, then the ``dual fusion
product" exists as a grading-restricted $\mathbb V^{\otimes M}$-module. Indeed,
we prove a more general version of this result without assuming $\mathbb V$ to
be $C_2$-cofinite. Our main method is a generalization of the propagation of
conformal blocks.
  In the special case that $\mathfrak X$ has genus $0$, when $N=2,M=1$, our
result is closely related to the theory of vertex tensor categories by
Huang-Lepowsky and Huang-Lepowsky-Zhang. When $N=1,M=2$, the dual fusion
product of $\mathbb W$ is closely related to Li's regular representations; if
moreover $\mathbb W=\mathbb V$, the higher level Zhu algebra $A_n(\mathbb V)$
arises naturally from the dual fusion product of $\mathbb V$.
