Let $\mathcal M_C$ denote the moduli space of rank two vector bundles over a smooth proper curve $C$ over an algebraically closed field $k$ of characteristic zero. Let $k(t)$ denote the function field of $\mathbb P^1_k$, and let $\overline{k(t)}$ denote a fixed algebraic closure of $k(t)$.
I would like to consider the direct limit $$\mathcal M:=\varinjlim_{k(t)\subseteq k(C)\subseteq\overline{k(t)}}\mathcal M_C$$ where the transition maps are the obvious pullback maps $\mathcal M_{C'}\to\mathcal M_C$ associated to maps of curves $C\to C'$ for $k(C')\subseteq k(C)$. Note that it doesn't matter that we started with $\mathbb P^1_k$; we could just as well have taken the direct limit over $k(C_0)\subseteq k(C)\subseteq\overline{k(C_0)}$ for any fixed curve $C_0/k$ and fixed algebraic closure of its function field, and it would give us the same moduli space $\mathcal M$.
Has this moduli space $\mathcal M$ been studied before? Can anything nontrivial/interesting be said about it?
I have been deliberately vague about in which sense I mean "moduli space of rank two vector bundles" (i.e. whether I mean a stack, or the GIT construction, or whether we impose stability conditions on our vector bundles, etc.), because I would be happy with an answer which says something interesting about any of these possible settings.
EDIT: I know a reasonable amount about each individual $\mathcal M_C$, and that others know even more. The point of this question is to say something nontrivial about the direct limit $\mathcal M$.
EDIT2: I am happy to restrict to bundles of trivial determinant, or, what is almost the same, to consider bundles $E$ and $E\otimes\mathcal L$ to be equivalent for any line bundle $\mathcal L$.
