[1] proves a lower bound for instances of mincost-flow whose bit-sizes are sufficiently large (but still linear) compared to the size of the graph, and furthermore proved that if one could show the same lower bound for inputs of sufficiently small bit-size it would imply $\mathsf{P} \neq \mathsf{NC}$ (and hence $\mathsf{P} \neq \mathsf{L}$). This is, at a high level, the same as Noam's answer in that it is about proving circuit depth lower bounds (=formula-size lower bounds), but seems to be a very different direction than Karchmer-Wigderson games.
In more detail, [1] shows the following. Using the same notation as in the paper, let $L$ denote the mincost-flow language. We can think of the mincost-flow language on $n$-vertex graphs, denoted $L(n)$, as a subset of $\mathbb{Z}^{k(n)}$ for some $k(n) = \Theta(n^2)$, with integers encoded by bit-strings. Let $B(a,n)$ denote the set of all vectors in $\mathbb{Z}^{k(n)}$ where each integer coordinate has bit-size at most $an$. Given a function $f(x_1, \dotsc, x_k)$ (we'll specify what kind of function later), we say that $f$ separates $L(n)$ within $B(a,n)$ if the points in $L(n) \cap B(a,n)$ are exactly those $\vec{x} \in B(a,n)$ such that $f(\vec{x}) = 1$.
Proposition [1, Proposition 7.3] If $L(n)$ is separated in $B(a,n)$ by $\det(M(\vec{x}))$ where $M$ is a matrix of size $\leq 2^{n/d}$ whose entries are (complex) linear combinations of $x_1, \dotsc, x_k$, and such that $a < 1/(2d)$, then $\mathsf{P} \neq \mathsf{NC}$.
The relationship between the bit-bound $an$ and the size bound $2^{n/d}$ is crucial here. In the same paper, he showed:
Theorem [1, Theorem 7.4] The hypothesis of the preceding proposition holds for all sufficiently large bit-bounds $a$.
The proof of the above theorem uses some heavy hammers as black boxes, but is otherwise elementary (note: "elementary" $\neq$ "easy"). Namely, it uses the Milnor-Thom bound on the number of connected components of a real semialgebraic variety (the same bound used by Ben-Or to prove lower bounds on Element Distinctness / Sorting in the real computation tree model), the Collins decomposition (used to prove effective quantifier elimination over $\mathbb{R}$), a general position argument, and few other ideas. However, all of these techniques only depended on the degree of the polynomials involved, and so cannot be used to prove $\mathsf{P} \neq \mathsf{NC}$ as in the above Proposition (indeed, [1, Prop. 7.5] constructs a polynomial $g$ of the same degree as $\det$ such that the above proposition fails with $g$ in place of $\det$). Analyzing this situation and looking for properties that went beyond degree was one of the inspirations for GCT.
[1] K. Mulmuley. Lower Bounds in a Parallel Model without Bit Operations. SIAM J. Comput., 28(4), 1460–1509, 1999
