This is a very interesting question and it is one that, as far as I know, has no perfect answer. An older answer would have something to do with cultural strength and power, and say something along the lines that strong cultures overpower weak ones even when defeated militarily. The problem with this argument, of course, is that it is circular: what is a strong culture? One that overpowers a weak culture! And so on.

If I had to give one generally applicable factor, it would be the needs of administration. Conquest is never on to a blank space, there are social and political relations that need to be responded to by the conqueror in one way or another. This is just as true of the Danish conquest of the Inuits as it is of the Mongol conquest of China. The cultural product of conquest is thus heavily dependent on how the conqueror and the conquered handle these relations. This can involve the conqueror subsuming themselves into the society of the conquered to better exploit the preexisting conditions, or it can involve the conqueror acting to destroy the earlier society. Both have numerous examples throughout history.

To give an example of this, consider four different conquests of Chinese territory by Steppe empires: the Khitan, the Jurchen, the Mongol and the Manchu. The Khitan only conquered sections of China and the Chinese never formed a demographically dominate category of their empire. The Khitan enthusiastically adopted the culture of the Chinese, particularly in artistic forms. The Jurchen later conquered the Khitan territory and eventually the whole of North China, meaning their empire was, from a demographic stand point, essentially Chinese. Perhaps because of this, the Jurchen purposely kept Chinese culture at arms length and jealously guarded their identity. The Mongols pursued a hybrid strategy, and in many ways their inability to handle the social contradictions of conquest led to their empire's collapse. The Manchu also pursued a hybrid strategy, but with more success: they maintained their distinctive identity, but operated more as an ethnic privileged class than conquerors, and did not fear Chinese culture of adminstration (as, for example, the Mongols did by importing foreign labor).

I would honestly doubt there is a universal law here, but these factors show up again and again.

Specifically with regard to language (which is just one facet of culture), there doesn't seem to be one general principle as to whether a conquering force will adopt the language of the conquered or vice versa. Reading "Empires of the Word: A Language History of the World," it seems as though individual circumstances involving conquest, religion, language prestige, economic factors (such as commerce) and other individual factors (such as the forced migrations of Aramaic speakers leading to it becoming a dominant language despite starting off as the language of a small group of seemingly unimportant people) play a large game of tug-of-war for influence in people's minds in a rather unpredictable way to determine which languages become dominant languages in a society. I imagine that something similar plays out in other facets of culture as well, but I don't have any evidence or source for that statement.