What happens to a cause after it has generated an effect?
I don't think it can cease to really exist, as it never really existed in the first place. Does the effect somehow - metaphorically - crowd out the cause? Or is the effect sufficient - even necessary - for the cause not to appear?
What is a good way of thinking about this?
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".
It's probably best to fragment this into two pieces for ease of comprehension.
Form-convention could be seen as the domain of cause and effect of which the mind is the cause and the body is the effect. Due to our clinging with the body we cannot see how the effect part is related to our actions; clinging to the body clouds our judgment and this is largely known as ignorance. In this instance, the causes are continuously put forth and the effects continue regardless - they just takes on different forms.
Formless wisdom is seeing this clearly to such a degree that all identification with the body ceases and the causes that were previously generated from body-clinging now cease.
In essence, all karmas converge at the mind, are dispensed by the mind and can be ended at the mind. There are other types of karmas which produce vipakas that linger like a fine mist. Not much you can do about those. When they come to the fore, manage them as wisely as possible.
So to answer to your question, the effects are always generated so long as we are identified with the body, but they take different forms. There can still exist residual vipakas (effects) that haven't yet come to light.
Your question is interesting. What happens to the cause...?? Simply put cause becomes the effect. That’s what happens. Take for example a bat hitting a ball. Bat hits and suffers a reaction due to the impact and a small dent is left on its surface. Similarly for ball, a small dent is left on its surface as well... another example would be putting ghee into the fire. Ghee burns flaring up the fire... ghee gets consumes and fire which consumes burns even more brightly...
I found a very helpful analysis from a respected academic.
"cause” is a label for the efficient discharge of a condition or set of conditions that effect a change. Once that moment of efficacy is discharged, the cause is gone, done.... to be communicative, [you] have to refer to a general category, not a specific particular. The category “cat” survives the life and death of all sorts of cats, since it is an abstraction. Likewise “cause.”
Furthermore, on the subject of whether a cause is just a cause, or can be something in addition to its causal discharge
According to most Buddhist schools, something is ONLY real (dravya) if it discharges causal efficacy. Some schools argued whether it only existed in that moment, or could have a latency period. Those accepting momentariness insisted on the first, the Sarvastivadins tried to make a case for the latter (and they had competing theories about how that would work).
Put differently, momentariness means that something ceases as soon as it has an effect: immediately; as soon as it arises.