I think the right way to think about this, according to Mahayana Buddhism, is that it is our mind that delineates the spatiotemporal continuum and identifies a cause.
In other words, things are not discrete, not discontinuous - so there are no separate "events" to begin with. In the general case, an event does not and cannot have a precise spatial and temporal boundary. Some events may have more or less well-defined boundaries but that's just a happy accident that we can't expect of most events. Usually when we examine an event very carefully we will see that it is "leaking" outside its boundaries, the boundaries are not accurate.
In summary, any "event" that we call "an effect" is a product of our own delineation. In Mahayana we say "event is empty", meaning it's an empty construct, not something concrete that exists ontologically but an abstraction we project onto our observation of the world.
Next, let's talk about the causes. Every "event" that we have thus delineated always exists in some context. The context is the environment that gave rise to the event. This environment is also contiguous and nondiscrete. As we analyze it with our mind we can identify some specific conditions that supported arising of our event. This is to say that an event does not usually (ever?) have a single cause. Instead, our mind delineates the background or the environment and identifies primary and secondary causes and conditions that were necessary for the arising of the event.
Now, what would happen if some of the secondary conditions did not exist or existed in a different way? Most likely the event they were to produce would be slightly (or very) different. Would we consider it the same event then? This is a very important question because it taps into the concept of "sameness" or identity. What makes an event itself? How different does it need to be before we start considering it an entirely different event? If we analyze this question deeply we will understand that sameness or identity is also empty, is also an abstraction, projection, convention, imputation.
From this last point we can conclude that so-called primary and secondary causes and conditions are also entirely our own designation. We pick some aspects of the environment from which an event arisen and designate them as important for this event, designate them as causes or even The Cause.
Finally, let's get to your actual question, what happens to the cause once the effect has arisen. Now we can understand that both the cause and the effect were our designations or our delineations to begin with. It's the entire environment that continuously and contiguously transforms.
The effect grows out of the environment like a sprout grows out of the seed (with other necessary conditions being the moisture, the warmth, the nutrients in the soil etc). What happens to the seed after the sprout has grown? Parts of the seed transform into the sprout while other parts dry or rot away. What happens to the molecules of water, nutrients etc.? They are used as the building blocks for the sprout. What happens to the warmth? The warmth or the vibration of the atoms is passed on to some chemical reactions that serve the growth of the sprout. Entire environment in the neighborhood of the sprout transforms.
Individual things only exist as analytical designations. Because everything keeps on changing (whether growing or falling apart or mixing with something else or gradually changing to something else) - in some sense exact things only exist for an infinitely short moment, and in the next moment they are no more, having passed their energy and information on to some successors (either slightly different from themselves or very different). In another sense things don't disappear at all, they just keep transforming and mixing with parts of other things. Neither of these descriptions is 100% accurate, but they point at Truth which is the living reality underneath our concepts.
This is the right way to think about this, as far as I know. If you want to do a formal study of this analysis, in Buddhism it's called "Madhyamaka".