Everett’s parameter includes a description of a world in which we exist, and all the records of that world’s history, from our memories, to fossils, to light reaching us from distant galaxies, exist. 

There will also be another universe exactly the same except that the “time step” has been advanced by, say, one second (or one hour, or one year). 

But there is no suggestion that any universe moves along from one time step to another. There will be a “me” in this second universe, described by the universal wave function, who has all the memories I have at the first instant, plus those corresponding to a further second (or hour, or year, or whatever). 

But it is impossible to say that these versions of “me” are the same person. Different time states can be ordered in terms of the events they describe, defining the difference between past and future, but they do not change from one state to another. All the states just exist. Time, in the way we are used to thinking of it, does not “flow” in Everett’s MWI.