That x is having a certain kind of conscious experience at t is not just a claim about what is happening right at t. If mental processes are in some way correlated with physical processes, then this follows from the fact that it does not really make sense to talk of the instantaneous state of the physical process (think, for instance, of wave phenomena or classical momenta—these are defined in terms of what happens at other times). But even without this correlation, this is plausible. Thought experiment: imagine seeing a red circle for a tenth of a second with no after image and no memory (the memory is wiped instantly). You see an obvious flash. Shorten the amount of time you're seeing the red circle. Eventually, you don't see it at all.
But if presentism is true, isn't this really weird? It would be really weird if my present conscious state were partly constituted by past-tensed states of affairs. The eternalist (or even growing block theorist) can talk of a temporally extended conscious state. That's not a problem. But the presentist can only talk of the present conscious state together with some (dodgy) past-tensed states, like having seen a red circle a quarter of a second ago. Of course, folks who think that beings coming out of swamps at random couldn't be conscious even if they had souls will not be bothered by this.