I have been musing upon the differences between process (process relational?) and object-oriented philosophy, as I have just encountered the contemporary versions of them. Here, I am thinking out loud and in public while trying to figure out what is meant by the terms, as I have been reading short guide after introductory guide. I stumbled upon process thought through Dewey and subsequently Peirce, and have discovered that I have been wholesale re-inventing much scholarship that had already been done. Hence, now I am playing catch-up.
From what I've gathered on some differences between "process-relational" and "object-oriented" philosophy, one is a difference in emphasis on what is ultimate and its status. I'll speak from my own Deweyan position that is encrusted with Peirce and years of mathematical training. For instance, what I get about the "object" is that it "withdraws" and maintains a hiddenness from its relations that is not reducible to them. This sort-of contrasts with a process view wherein relations are internally constitutive such that to be is to be related (else we can say nothing of it, which is to say that it doesn't exist, and we have little to say about modalities of being that do not exist and are not related), although there is an ontological difference between an the emergent actualization and its conditions. (I believe this is called an actual occasion in Whitehead, yet I lack sufficient mastery to assert that.) What confuses me about the difference is that I do not see the basis to claim that an object "withdraws" from its relations unless one presumes that the object has some special ontological status not granted by its constitutive conditions. While I am will to allow emergence (~creativity) and epistemic limitation, I cannot help but think that this view is vestigial substance metaphysics. That is, there remains a wish for a fundamental unit of determinate identity, the object, whereas a relational/temporal/historical view of (individuated) identity spreads it out. For example, a relational identity can be designated in terms of an atemporal function (not recommended as isomorphisms will predominate), or in terms of ordered transformation (less problematic) that requires a historical concept of identity. The latter could be described as a telos, but I don't see how any version of teleology would work unless it were Aristotelian, in which case teloi are directly causal. (If what I mean isn't obvious, this will take some explaining of in what way teloi are causal vs. emergent.) Regardless, the alternatives that immediately come to mind for identity all destroy anything like an "essential" conception of identity, which is what I presume is wanted of the "object" in object-oriented.
In conclusion, my hunch about object-oriented philosophy is that there's a want for determinate identity or some determinate basic ontological unit.