> I'm not even sure why focus on casuality of the event.

Because your original question made it seem like you were thinking of the density decrease caused the volume increase. If you understand that that's not how it works, that's good.

> Can we imagine a noticeable area of space suddently becoming empty and devoid of matter

Note that this can't happen except by matter flowing out of the area. The matter can't just disappear. That is what the local conservation law for stress-energy says.

> would it cause the inflation, as in elongation of paths between ourselves and some distant object with this area between us, to become faster? Slower? Just about the same?

As I understand it, cosmologists are working on models in which the density of matter does vary from place to place as well as with time (although even phrasing it that way introduces technical issues that I won't go into here), because our observations seem to indicate that, for example, there is a "void" a billion light-years or more wide in our general vicinity, where the density is significantly lower. Note that this only applies to ordinary matter, not dark energy.

The general indications so far of those models, from what I understand, is that such "voids" result in a faster expansion in that area (though again I am ignoring significant technical issues with how this is phrased). However, this is still an open area of research, and the models become significantly more difficult to work with because you can't solve the equations analytically any more, the way you can when the density is constant everywhere in space. You have to do numerical simulations.

> When you talk about a "local conservation law", do you mean that there's math which makes the energy difference go away, and then inflation goes at the same rate regardless of whether energy-losing matter (such as light) is present in the expanding area or not?

I'm not sure what this means.

What the local conservation law says is that stress-energy cannot be created or destroyed; it can change form (for example, matter can transfer some of its stress-energy to radiation by emitting it), and it can flow between regions of spacetime, but the flow cannot have any sources or sinks. The complication is that the "flow" has to take into account the spacetime geometry, which affects how you "count" the flow. For example, if we consider an expanding universe containing just ordinary matter, the energy density decreases like the cube of the scale factor, but that is not a violation of the local conservation law, it's because of the local conservation law--the spacetime geometry is changing as well as the energy density, and the combination of the two changes keeps the local conservation law satisfied--in terms of the local conservation law, the "flow" is conserved, there are no sources or sinks, even though the energy density is changing.

This is one of those things that's really, really hard to explain in ordinary language. The mathematical statement is that the covariant divergence of the stress-energy tensor is zero. Physicists understand what that means and how to use it even if they can't express it very compactly in ordinary language.