So, a basic set-up in modern enumerative geometry is that you have some object $X$ (say, a "nice" stack, for whatever definition of "nice" you need) and then you want to "count" the curve of genus $g$ intersecting a bunch of cohomology classes and of degree $\beta$, so you look at $\bar{\mathcal{M}}_{g,n}(X,\beta)$, then pull back the classes, intersect, and get a Gromov-witten Invariant. Famously, this gives the Kontsevich formula counting rational curves in the projective plane passing through $3d-1$ points. And though GW-invariants can be negative and rational, there are nice cases where they do count something legitimate, such as the genus $0$, $n\geq 3$ case into a homogeneous space.

So, enough background, here's my question (and this is largely idle curiosity, so no specific motivation): can we do this in higher dimensions? For instance, given a (smooth?) variety $V$ and marking a bunch of subvarieties $W_1,\ldots,W_n$ (maybe restricting them to points?) can we form $\bar{\mathcal{M}}_{V,(W_1,\ldots,W_n)}(X,\beta)$ a moduli space of stable mappings of varieties deformation equivalent to the one we started with into our space, represented by a given cohomology class $\beta$ and with $W_1,\ldots,W_n$ intersecting some cohomology classes, so that we can get something that can be called higher dimensional Gromov-Witten invariants? If this has been studied, under what conditions does it actually count subvarieties? For instance, if $X$ is $\mathbb{P}^N$ and $V=\mathbb{P}^2$, and maybe if we loosen things to just needing rational maps, could we use something like this to count rational surfaces, satisfying some incidence conditions?