There is a lot to be said about Marxist and Marxian approaches to history – for a distinction see this thread and the answer by /u/tenminutehistory – and the shortcomings and critiques of the former. Which is why it is best you check out this answer by /u/ThucydidesWasAwesome going into the subject matter including the 'Asiatic Mode of Production' about which they write:

There are many serious critiques of Marxism. An easy one that I cite often on this sub is Marx' position on the 'Asiatic Mode of Production' (a catch-all for non-European socio-economic systems) which was eventually torn apart by many authors, including Marxist historian Perry Anderson's devastating essay on it included as an addendum in Lineages of the Absolutist State (Verso, 1974). I should note, however, that as Anderson himself shows, this was a barely formed idea that Marx' followers frankensteined together from numerous mutually contradicting scribblings over the years and not a fully developed idea that was central to dialectical materialism at large. Other critiques may focus on the intentionality of the approach, which isn't the reproduction history "Wie es eigentlich gewesen" (history as it really was, as Von Ranke put it), but rather a critical analysis of history always looking to point out the ways in which a given social system exploited some to the benefit of others. Remember, Marx is creating this theory as one explicitly meant as a critical tool to understand the plight of the poor, reveal exploitation which has otherwise been minimized, and thus imbue the exploited with an understanding of their reality which will compel them to change society. Another common critique is that while in theory Marx' approach as supposed to be dynamic, alive, capable of change, etc., in practice his writings became the object of quasi-religious reverence from his followers who preferred to cite them as revealed truth. This critique isn't entirely untrue, especially in much of the socialist world during the Cold War where Marx' writings became a matter of dogma. However, these kinds of 'vulgar' Marxism (quasi-religious, poorly understood or applied, etc.) aren't representative of all Marxism, as you have a great many excellent Marxist historians who put out a great deal of original work which was not dogmatic or unchanging at all. English Marxist historiography is an excellent example, having produced many historians of the first order, including Perry Anderson and Eric Hobsbawm. There are other critiques tied into his understanding of economics and the legacy of these supposed misunderstandings in the work of those who follow his ideas, but that's a 5-10 page essay right there and I am not an economist myself (I will note that many of these critics don't seem to understand Marx' ideas either so I have yet to be fully convinced by them on most points). The examples in this paragraph aren't an exhaustive list of all the critiques of Marxist historiography, which are as numerous and nuanced as the critics themselves.

But if you are looking for Marxian analysis and how it can be applied to history quite fruitfully, I'd suggest checking out this post on Gramscian Hegemony and history or this one about Marxian approaches to the Holocaust.

I'm less than familiar with Marxist analysis of the French Revolution and the place of these interpretations within historiography. However, the little I know does jive with the idea that classical Marxist class analysis of the Revolution have fallen out of favor.

As I argue in the thread cited by /u/commiespaceinvader, I agree that Marx himself was Eurocentric, however I disagree with the statement that dialectical materialism itself is Eurocentric.

The Asiatic Mode of Production reveals both ignorance of non-European societies and many of the prejudices common to Marx' time and place (19th century Central and Western Europe). The central ideas of dialectical materialism (focus on the economy, a study of social conflict in society in terms of material interests, an attempt to make history a social science, etc.) are both universal and, for the most part, common sense at this point.

In a time and place where history could often be a parade of names and dates of kings and battles, Marx tried to create a coherent methodology which would allow for a scientific interpretation of reality and, potentially, the ability to predict the outcome of social conflict.

Was he successful? Not really. As I've said elsewhere, Marx' theories on historical analysis are deeply flawed and many of his concept (like Mode of Production) have either changed a lot or been discarded completely.

But for Marx' method, that was the point. He wasn't trying to say 'okay, here is a method, the method, to figure things out. I am the final prophet of the social sciences. After me there shall be no others.' Instead he as offering tools, of his creation or which he adapted from others (like Hegelian dialectics), in the hopes that his theories would be useful and push others to develop their own hypotheses.

Half the problem of super-orthodox Marxist analyses is precisely when they frankenstein together theories which Marx himself only half thought out, then try to apply them or affirm them as a matter of fact.

Marx' ideas are a product of 19th century Europe. Wherever they've survived, they've evolved. A critical methodology that took the side of the oppressed, instead of a superficial methodology which simply reproduced existing structures of power, is a project which has been emulated (with varying degrees of success) by many disciplines within the social sciences. The idea that history is a social science at all, with its own method and theory, has also become common sense. The idea that class conflict is an important, though not always the central, part of a historical process is also deeply rooted in academia today.

Think of it as Romaness. The Western Roman Empire fell in the 5th Century AD, but the Eastern Roman Empire (often called the Byzantine Empire) survived until the 15th. Maybe a Roman of the time of Julius Caesar wouldn't have recognized Byzantines of the 15th century as 'Roman'. That's because Byzantium continued to exist and evolve. It would have been stranger, indeed, if Roman culture (be it in the East or the West) had stayed static.

The same is true for Marxism. Marx is dead and his ideas have evolved since his passing, but Marxist influence is alive and well.