I suspect we disagree on less than you think. This is a complex equation but, despite continuities, the successor states (especially outside of Mongolia and parts of Central Asia) saw strong reasons to assert what made them distinctive alongside any shared values/institutions.
I think we do disagree, though. As I'm concerned, I can see no cultural nor religious transformations that would prevent Mongol unity and make elite networks, even more so branches of the imperial family, growing apart, and thus being an explanation for the decline of the Mongol Empire.
Neither Islamization not Turkification, which are the two main phenomenona I can think of, would have prevented individuals and groups to still identify themselves as part of the Mongol political construction.
Of course, regional networks, loyalties, and identities would have developed. But are they truly cultural, or rather political? I would really like to know what fundamental cultural differences divided the Mongol elites on either side of the Caucasus or the Oxus.
I would argue that, beyond these regional political affiliations, the elites, and even more so the members of my imperial family, shared a whole set of common cultural, political, ideological, and even religious references, which bound them deeply.
As late as 1405, Temür still aknowledged the Qubilaid qa'ans as the supreme overlords of the Mongol world when he ordered his last great campaign towards Mongolia —and not China, as often thought— to put things in order there and install his own candidate on the throne.
I would go even further. We tend to essentialize the post-1259 partition too much as a division of the empire into four distinct entities, whereas it was in fact a reconfiguration and decentralization of imperial power along lines more complex than the simple “four ulusud,”...
and without necessarily challenging the imperial unity. That is why I find it more relevant to look for the common causes of the decline of the Empire as a whole, which was affected in its entirety by the crisis of the 14th century.
We may have to agree to disagree. Culture, identity, and politics were and have always been closely woven in my view, for the Mongols as for so many peoples. Shifts in religious identity are especially profound - that's why we hear so much about them in the sources, ...
even when we learn very little about so many other things. Authors (often writing on behalf of political patrons) considered these matters important and worthy of record. I'm not sure how much further we can take our discussion given that we do seem to fundamentally ...
...disagree on this point, but I've enjoyed our conversation and may I wish you the best is your research.