It would be uniform.
The crust of the Moon is, on average, about 50 km thick. There is indeed some stratification in its composition, with upper layers composed largely of feldspar and a lower layer composed of more mafic rock (Arai et al. 2008). The upper feldspathic layers appear to be $\sim$10 km thick, meaning that the composition at the bottom of your 2 km-deep hole will be fairly similar to that at the top. You'd have to make it about 5-10 times as deep to see significant composition changes - and even at that point, you'd still be far from the mantle!
That said, the properties of those rocks would change as you went down - though it's not clear how drastically. For example, it might also be interesting to consider the subsurface temperature gradient, although that's not well-known throughout the entire crust. To at least 20 meters below the surface, the gradient is roughly 1-2 Kelvin per meter (Krotikov & Troitski 1964), presumably slowing down considerably after that (possible as low as $\sim$1 Kelvin per kilometer at some point). You'd definitely see some sort of significant temperature differential immediately after the hole was finished, although I'm not sure what effects that would have. There would also be increases in density and pressure with depth.
(I should note, of course, that the crust on the far side of the Moon and the crust on the near side of the Moon differ slightly in thickness and depth, but not enough for this answer to be particularly location-dependent.)