Perdurantism holds that we either are four-dimensional entities, stretching like worms through spacetime, or temporal stages of these. Most worm-theorists also believe in temporal stages—they just do not identify us with the stages, but with the whole. (I myself am attracted to stageless worm theory.)
Now here is something that strikes me as a bit interesting. The founding intuition of four-dimensionalist theories is that there is a very close analogy between space and time, and four-dimensionalists pay a lot of attention to science. But, then, positing stages goes against the spirit of four-dimensionalism. For if we pay heed to the space-time analogy intuition, we would have to posit something analogous to temporal stages, stretched out through time the way temporal stages are stretched out through space. While temporal stages are three-dimensional (maybe with a bit of thickening), spatial stages would be one-dimensional (maybe with a bit of thickening), stretched out through time—basically cores cut from the worm along timelike curves. And just as the four-dimensional worm is made out of temporal stages, we would have to think of it as made out of the spatial stages.
But that would be mistaken. The only plausible way in which we could be composed of spatial stages would be if we were made of particles, and we could trace the trajectory of each particle through spacetime. But even if there are particles, the uncertainty principle makes the possibility of such tracing dubious. And quite possibly, physics will dispense with localized particles altogether.