This is a pretty complex theoretical question that does not really have any straightforward answer--or rather it does have one, a very simple one, but that one opens up a lot of further discussion. Because of that I am going to try to respond to this in a fairly compressed way, but would be happy to follow up as much as I can if I mention anything that seems to strike your interest. Fundamentally, I think there are three ways to think about this question: a glib, quick response, a theoretical grounding on how a city functions within a larger political economy, and an empirical examination of how cities in practice did die.

The glib response is that it is very easy for cities to die and very difficult to keep cities alive because people need need to eat and a pretty base requirement for a city is that it consumes more primary products than it produces. If the economic or political factors that maintain the constant flow of supplies into a city from its hinterland end, the city will also end.

The hinterland is a central component of this, broadly defined as the region from which a city gets the fundamental material it requires for its functioning. So if you think back to a city like Ur, it was certainly part of trade networks that spanned extremely large distances, but its immediate hinterland would be quite limited by modern terms as the nearby areas its received its food from. If you jump ahead to Rome at the height of the empire, it depended for its day to day functioning on lines of trade and communication that spanned the entire Mediterranean Basin. If it was bigger than any city before it ever really was, it was because it could draw upon the resources of a wider hinterland than any previous city. If we jump ahead again to something like modern New York I don't even know what the "hinterland" would be--maybe the world (alternatively, in a modern, developed economy, perhaps the "resource" would be people, although I am zigging a bit out of my zag there). But fundamentally, you cannot have a city without a hinterland. I think there used to be a bit more consciousness of this, William J Bryant's "Cross of Gold" speech noted that "Burn down your cities and leave our farms, and your cities will spring up again as if by magic. But destroy our farms and the grass will grow in the streets of every city in the country."--but that is also a out of my zag.

Despite that bare fact, the relationship between a city and a hinterland can vary a great deal based on historical, political and economic circumstances. One of my favorite works of sociology, The City by Max Weber which is a really great read if anything in this seems interesting, divides the character of of this relationship into three "ideal type"^1 cities: Consumer cities, producer cities, and merchant cities. A "consumer city" exists due to its position within the geography of political power. A classic "consumer city" would be one where a king or elite lived who subsisted off of taxation and rents extracted from the hinterland. Rome would be a classic example, as its subsistence was fundamentally dependent on the power of the elite to extract resources from the Mediterranean. In a very different form, one can think of Washington DC--which owes its existence due to the presence of the federal US government. A "producer city" exists because its "economic versatility" (multiplicity of economic possibilities that exist due its existence)can return value to the countryside. In Weber's telling, the essentially non-urban character of Medieval elite means that the towns that did develop took this form, hence the relative importance of craft guilds. The merchant city is more or less what it sounds like--its existence is owed to its location within the geography of trade (although looking at the essay again he seems to fold this concept into "producer city"). But in all of these cases, the fundamental question of a city is how it gets what it needs to survive.

In real world terms, the fall of the Roman Empire provides quite a few examples of how cities collapse and fade away. A really great book on this is Bryan Ward-Perkins' Fall of Rome and End of Civilization (which has received some critique but is grounded in the debates in Roman archaeology around economic history and so I think it is a good choice for this topi--also, widely available and very readable). The fall of the western empire (which ward Perkins' placed around 400-600) resulted in the breakdown of chains of connection linking the various regions together. For the most obvious example, in the preceding centuries the city of Rome itself received large and consistent quantities of grain from Sicily, Egypt and North Africa. The political and military turmoil of the period made these trade routes no longer viable, and particularly when the Vandals conquered North Africa in the 450s the city of Rome went into a tailspin. In 150 CE the city had perhaps a million inhabitants, in 350 perhaps half that, and in the 600s was in the tens of thousands. Granted, it still existed as a city so did not "die", but it suffered enormous population destruction because its relation to its hinterland was radically and rapidly altered. Many cities were not even as lucky as that: London had perhaps 40,000 people in 300 CE, but by 500 was essentially abandoned. The complete collapse of the Roman political system in Britain essentially de-urbanized the entire city. While part of a somewhat different process, Berenike is another useful example. A port along the Red Sea, it became a small but permanent city during the third century, but as the trade routes between India and Rome dried up during the end of the classical period, it did as well, and it was abandoned.

There is not really any one way a city might die, but a city can only exist through resources it obtains from its hinterlands, however it does so. If its capacity to receive those resources is ended, so too will the city.

^1 an "ideal type" is a useful framework to think about a given social topic that does not necessarily match any particular example perfectly.

While I'm not sure I'm capable of fully answering your question I did make a post a while back talking about the decline in southern Mesopotamian urban settlement in the centuries leading up to the Common Era. It touches on some of the reasons that an ancient urban settlement might decline to the point of abandonment.