The perdurantist thinks that we are four-dimensional beings made up of three-dimensional slices, temporal parts, from which we inherit our changing properties such as thinking. One good reason to deny perdurance is that implies that our thinking is derivative from another entity's thinking, namely from the part's thinking, pace Andrew Bailey's very plausible thesis that our thinking does not derive from another entity's thinking. Another issue is that perdurance has at most a 50% chance of being true for me: since the slice thinks the same thoughts as the four-dimensional being, I have at least a 50% chance of turning out to be the slice--contrary to perdurance.
But there is an interesting Aristotelian version of perdurance. I am a four-dimensional being, but I have a sequence of special accidents Dt corresponding to the times t at which I exist. Then all my changing features are grounded in features of these accidents. For instance, I am thinking at t provided that Dt is thinking*, where thinking* is whatever feature of an accident Dt that makes the possessor of Dt be thinking. For categorial reasons, thinking* isn't thinking: only substances think, but non-divine substances think in virtue of having an accident that in turn is thinking*.
What are the Dt accidents? One option is that they are the accident of existing at t. But perhaps there is a more Thomistic option: perhaps in the case of material substances they can be identified with something like Thomas's accidents of dimensive quantity. Thomas thought that material substances had a special accident, a dimensive quantity, and all their other accidents were in turn accidents of its dimensive quantity. This is a very similar role to that played by Dt. Or, perhaps, we could take Dt to be an accident of occupying such-and-such a three-dimensional region of four-dimensional space. There is room for further research here (and if anybody wants to work more out and co-author, they are very welcome).
There is a major difference in outlook between this and typical perdurance pictures. On typical perdurance views, the slices are prior to the four-dimensional whole. On this Aristotelian perdurantism, the Dt accidents are, like all accidents, posterior to the substance, which is four-dimensional. Apart from this, the view might not be that distant from standard perdurantism. I have proposed in another post that an Aristotelian could identify parts with certain kinds of accidents. On that identification, the Dt accidents could turn out to be parts. But the difference in outlook remains: the parts really are just accidents of the whole. And the parts don't have the same features as the whole does. They have features for which we have no names, features we can only identify as that feature of the accident that grounds the substance as being F.