A multi(co)span is supposed to be something that generalizes span (and cospan) both horizontally and vertically: it may have a number of legs different from 2, but more importantly it need not be a single roof but can be a more complex diagram.
A multispan is supposed to be a model for a hierarchical cell complex with
a single top “cell” $K(\top) \in C$ (an object in some ambient category $C$);
for each cell $K(a)$ a collection of morphisms $\{K(b_i) \to K(a)\}_i$ into it, to be thought of as pieces of boundary components of $K(a)$;
and so on;
such that the entire resulting diagram commutes, expressing the fact how one boundary pieces may be part of different higher order boundary pieces.
Multi-cospans of cubical shape have been introduced and studied by Marco Grandis in his work on Cospans in Algebraic Topology. Grandis also formulates the idea that an extended QFT should be a morphism with domain extended cobordisms modeled as multi-cospans.
Special cases of multispans of the above general kind are
ordinary spans and cospans: these are obtained by restricting the domain poset to be of the form $\wedge := a_1 \to \top \leftarrow a_2$.
the double cospans of the form $\wedge^{\times 2}$ with $\wedge$ as above, described in
Grandis’ higher cubical cospans, which generalize the previous example: these are obtained by restricting the domain posets to be the cartesian powers $\wedge^{\times n}$ (with $\wedge$ as above).
An attempt at a discussion of multispans in greater generality is at hyperstructure.