Theory of momentariness -- do things arise ex nihilo? As far as I understand it, according to the theory of momentariness (Kshana- Bhanga, or kṣaṇa) the cause does not exist in the effect. A perishes before B arises. It arises from nothing. Even if we take the Zero energy universe model, shunyata is seen as emptiness of emptiness, and the Zero energy universe is more akin to Nirguna brahman than Shunyata. Nothing arises from nothing.

My question is -- even if we say that the cause creates the effect, with what material does it do so? For example, if A gives rise to B, where did B come from? Is it the same substance/thing as A in a different form? If not, how did B arise from A? Even if A exists after or until B arises, how can a new B arise from A (unless B is A in a different form like water turning into vapor or unless A divides its substance/material into different parts)?

Things cannot arise ex nihilo can they? If we assume that things are created anew, that is actually ex nihilo (and is also a problem that many Theists have with their God creating things from nothing).

In actuality new things cannot be created, we see one root substance being transmutated into different components and forms but the root substance remains the same. Things cannot infinitely divide themselves. A cannot divide itself to become B, because the material would become infinitely smaller.
 No new materials are created. From what would they be created by? Actually new things in actuality cannot arise from existent things, let alone when such existent things have already perished before their effect arises, as in the case of khshana-bhanga. In nature we just see one substance being transmutated into different forms but the root substance remains the same.

I think that materialism makes more sense than the theory of momentariness for this reason?
The theory of atoms perishing and then giving birth to new atoms makes no sense. Even if Atoms didn't perish until new atoms were created, this would still be impossible as new things cannot arise substantially or materially as there is only transmutation/transformation in nature.in materialism that would be eternal subatomic components.
This is essentially the question: "Does X exist?"; "Does X not exist?" which are questions the Buddha puts aside as not being helpful towards achieving the goal. Whether or not X exists, there is pain (dukkha). This is the problem that is dealt with in the Buddha's system.

The idea of 'cause', as 'hetu' and 'nidana' and 'paccaya' and a half dozen other terms are translated (including 'ahara' food!), is an error. "Hetu" the most reasonable candidate for 'cause' means 'driving force' (something like the teamster's 'giddy-up!'). Cause remains a mystery that is not dealt with as it is irrelevant to the solution of the problem of pain.

What we get in the Paticca Samuppada (and other formulas dealing with the origin and ending of things) is not a description of cause and effect (no matter how often it is translated that way -- it makes no sense as a cause-and-effect description) it is a description of causal association (aka: Bhk. Thanissaro: requisite condition; aka: proximate cause, economic cause). 

Causal association can be seen. 

That is the point.

We deal in this system with what we can see and know for ourselves; what we can do something about ourselves.

"It cannot exist without it" does not equal "cause."

We cannot see the 'cause' for the appearance of aging and death from birth, but we can see that without birth there would be no aging and death. 

So without needing to deal with 'cause', we can know for certain that the result, whatever the 'cause' may be, can be eliminated by eliminating birth. And so on for the other causal associations.